This HTML document contains all the results and plots
supporting the paper network structure shapes languages:
disentangling the factors driving variation in communicative
agents. All the data and scripts needed to reproduce this
document are available in the GitHub
repository.
This HTML document uses the following font and color
conventions:
fixed font textThe full information about the version of R (R Core Team, 2022). The packages used to obtain
this html file can be seen when opening the corresponding
Rmarkdown file.
To perform the simulations, we used two different machines:
For more information on the model, refer to the main paper.
We conducted three types of simulations:
DATASET 1. In the first, we systematically compared two sets of networks differing in exactly one metric.
DATASET 2. In the second, we used other datasets (with automatically generated random, scale-free, and small-world networks using a wide range of parameters) to complement these results and look at:
DATASET 3. In the third, we looked at the language value and variation in heterogeneous populations (where some agents are locally biased) in all types of networks.
The Netlogo program is available in the GitHub repository
associated to this Supplementary Materials (see my github under
the name Netlogo_program_dirichlet.nlogo. This is an
extension of the Netlogo program available in our previous paper “Interindividual
variation refuses to go away: a Bayesian computer model of language
change in communicative networks” (2021) published in Frontiers
in Psychology, and available in the Supplementary Materials of this
paper here.
While the model is the same, a few functions have been added (for
example, it is now possible to study multinomial language features).
To understand how to use our Netlogo code and parameters, please
refer to the Netlogo guide, a pdf file that can be found in the same github
folder (see appendix_netlogo.pdf file)
Using this Netlogo file, we generated networks with specific
properties. The results of the simulations are stored in a
Raw folder. They have the following name:
DN_typesimulation_networktype:
N can take the values 1, 2, or
3 depending on the dataset numbertypesimulation can take the values continuous
or multinomial depending on the language studiednetworktype can take the values scalefree,
smallworld, or random based on the network used to
generate the simulations.For example, a file D1_dirichlet_random.csv present in
the folder Raw is the raw file obtained from Netlogo (using
the code called D1_dir_random in Behavioral Space), and
includes data obtained using random networks, a
multinomial language, for the dataset 1 (comparing
sets of networks).
Below are the specific parameters used to generate networks on Netlogo.
In Dataset 1, we conducted 100 replications for each combination of parameters. In Dataset 2, we generated 900 networks for each network type. This entailed 10 replications per combination for random networks, 5 replications per combination for small-world networks, and 100 replications per combination for scale-free networks. This choice was guided by the aim to maintain models with a similar amount of data. However, it is important to note that we also tried to study models with 100 replications for each combination of parameters within each network type (so much more data for small-world compared to scale-free networks), but this change did not impact any of our conclusions.
For scale-free networks:
| Dataset number | Size network | Init value Dirichlet |
|---|---|---|
| Dataset 1 | 50; 100 | 0; 5; 20 |
| Dataset 2 | 50; 150; 300 | 0; 5; 20 |
| Dataset 3 | 150 | 0; 5; 20 |
For random networks:
| Dataset number | Size network | Connection probability | Init value Dirichlet |
|---|---|---|---|
| Dataset 1 | 50; 100 | 0.06; 0.24; 0.2;5 0.26 | 0; 5; 20 |
| Dataset 2 | 50; 150; 300 | 0.05; 0.15; 0.25; 0.35; 0.45; 0.55; 0.65; 0.75; 0.85; 0.95 | 0; 5; 20 |
| Dataset 3 | 150 | 0.08 | 0; 5; 20 |
For small-world networks:
| Dataset number | Size network | Number neighbors | Rewiring probability | Init value Dirichlet |
|---|---|---|---|---|
| Dataset 1 | 150 | 24 | 0.1; 0.9 | 0; 5; 20 |
| Dataset 2 | 50 150 | 2; 4; 8; 24 | 0.1; 0.3; 0.5; 0.7; 0.9 | 0 5 20 |
| Dataset 3 | 150 | 2 | 0.1; 0.9 | 0; 5; 20 |
The value for initial language exposure (which we refer here as init_lang_exp) tells the number of utterance that have been heard after the agent is born. Since the agents are born with a vector = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], then their value in \(u_4\) becomes 6 (5+1) or 21 (20+1).
In the datasets, this variable has three modalities:
In these Supplementary Materials, we preferentially refer to this variable with the modalities “Without”, “With (weak)”, and “With (strong)” (instead of language change and emergence) because it allows us to contrast between the two language change conditions (weak: \(u_4\)=6 and strong: \(u_4\)=21).
After we performed the analysis in Netlogo, we obtained files which
often contain thousands of simulations. Thus, these files are very heavy
and hard to read, containing many unecessary informations (useless
column, etc). (For example, for each network, we stored the final value
of the Dirichlet vector for all agents!). We used another
Rmarkdown file to clean this data (see
CleanData.RmD in the GitHub
folder). This file first selects only the columns containing meaningful
information. Then, it computes the following:
first, it computes the exponent of the scale-free distribution given its degree distribution (thus, this step was only performed in scale-free networks). This exponent is what we will call the variable “degree distribution” (important note: we called it so for simplicity reasons, but this variable does not actually refer to the degree distribution! it shows the exponent, which is an indicator of the shape of the degree distribution). All values are usually included between 2 and 3 as the scale-free networks were generated using Barabasi-Albert algorithm. However, networks with an exponent of 2 have a distribution less skewed compared to network with an exponent of 3.
second, it shows the density plots for the intersection of our different metrics (pathlength, assortativity, size, clustering coefficient, degree distribution): for example, we plot the values of the different simulations of pathlength in the x axis and the value of assortativity in the y axis. Thanks to these plots, we could visualize what values of the different metrics to select while keeping one constant. Then, we selected sets of 100 networks with the desired values (see the method in the main paper for more information). Please note that we used several types of networks to investigate the effect of all metrics. Indeed, it is not that easy to find networks where only one metric differs from the others! Thus, we used scale-free network to investigate the effects of pathlength, assortativity, size, and degree distribution, random network to disentangle the effects of mean degree and pathlength, and small-world network to investigate the effect of clustering coefficient.
For example: for a network with a size of 50 people and an initial
language exposure = with (weak), here is what looks like the plot with
pathlength in x and assortiativty in y (one point is the average value
for one network).
To determine the thickness between the two black lines and the diameter of the red circles, we manually adjusted the values until we identified the smallest possible differences between our sets. It’s important to note that other values could also be suitable. We experimented with various alternatives, such as smaller values for the control variables, resulting in smaller differences in the variables of interest. However, these adjustments did not alter the main conclusions.
third, it computes the mean of each metric for
each condition. This data is gathered in a table, which is then exported
under a file ending by _tab_merge.csv.
fourth, we checked the maximum language value
for each participant, for all replication. We save this dataframe in
files ending by _lang_merge.csv.
fifth, we computed the measure of
inter-individual variability (by estimating the Kullback-Leibler
divergence on all pairs of agents) and intra-individual variability (by
estimating the entropy for all agents, and then applying the mean or the
standard deviation). We save this dataframe in file ending by
_merge.csv. Please note that the computation of
Kullback-Leibler divergence is quite long.
All these datasets were exported in a folder called
Multinomial for simulations that used a multinomial
language, and in a Continuous folder for simulations that
used a continuous language.
first, it computes the exponent of the scale-free distribution given its degree distribution.
second, we computed the measure of
inter-individual variability (by estimating the Kullback-Leibler
divergence on all pairs of agents) and intra-individual variability (by
estimating the entropy for all agents, and then applying the mean or the
standard deviation). We save this dataframe in files ending by
_merge.csv (note that the computation of Kullback-Leibler
divergence is quite long).
In these datasets, we did not manually selected sets of networks.
Then, we aggregated these datasets into the same files. These files
are the one used in this analysis and are available in the
Summary folder:
| Input file | Dataset | Type simulations | Type network | Information type | |
|---|---|---|---|---|---|
TIMELINE_continuous_scalefree.csv |
Timeline | continuous language | scale-free | show the variation and language at each round in a few populations | |
TIMELINE_dirichlet_scalefree.csv |
Timeline | dirichlet language | scale-free | show the variation and language at each round in a few populations | |
DATASET1_continuous_table.csv |
Dataset 1 | continuous language | scale-free, small-world, random | gather information about the metrics values for each set in a table | |
DATASET1_continuous_variation.csv |
Dataset 1 | continuous language | scale-free, small-world, random | gather information about the language and variation (inter and intra) for each simulation | |
DATASET1_dirichlet_table.csv |
Dataset 1 | dirichlet language | scale-free, small-world, random | gather information about the metrics values for each set in a table | |
DATASET1_dirichlet_variation.csv |
Dataset 1 | dirichlet language | scale-free, small-world, random | gather information about the variation (inter and intra) for each simulation | |
DATASET1_dirichlet_language.csv |
Dataset 1 | dirichlet language | scale-free, small-world, random | gather information about the language for each agent in each simulation | |
DATASET2_dirichlet.csv |
Dataset 2 | dirichlet language | scale-free, small-world (with different rewire probability and average number of neighbors), random (with different connection probability) | gather information about the variation (inter and intra) for each simulation | |
DATASET3_dirichlet_variation.csv |
Dataset 3 | dirichlet language | scale-free, small-world (with two differnt probability of rewire), random | gather information about the variation (inter and intra) for each simulation | |
DATASET3_dirichlet_language.csv |
Dataset 3 | dirichlet language | scale-free, small-world (with two differnt probability of rewire), random | gather information about the language for each simulation | |
DATASET3_dirichlet_language_2.csv |
Dataset 3 | dirichlet language | scale-free, small-world (with two differnt probability of rewire), random | gather information about the language (different type of language data) for each simulation | |
DATASET3_supplementary.csv |
Dataset 3 | dirichlet language | scale-free, small-world (with two differnt probability of rewire), random | structural information about the local metrics when the network is first build (no evolution of language with time) |
Before looking at the results from the three datasets, we perform a preliminary analysis aiming at understanding when we should stop the simulations.
As a reminder, here, time is discretized into iterations, starting with iteration 0 (the initial condition of the simulation) in increments of 1. At each new iteration, i > 0, all agents produce one utterance using their own internal representation of language and production mechanism (as described in the method section of the main paper). Thus, the language of the population evolves at each iterations. We first aim to look how to the language evolves over the course of iterations. This allows us to choose after which iterations we stop the simulations.
The stabilization time was already explored in our previous paper (Josserand, Allassonnière-Tang, Pellegrino, & Dediu, 2021), thus we did not go into detail in this process here. Stabilization time captures how long (in terms of interaction cycles) it takes for the language of a given network to reach a stable state. In our previous paper, we estimated the stabilization time based on the method developed in (Jannsen (2018)) (p. 79). To do so, we used a fixed-size sliding window within which we estimate the change in the language value, we multiply this number by 10,000, round it, and stop if this number is equal to zero (i.e., the slope is within ±0.00001 of 0.0) for 50 consecutive steps. Practically speaking, the maximum number of ticks of our model is nIterations = 5000, and the size of the sliding window is \(W = nIterations/10\). For a given window, we estimated the change, \(t(e_g)\) using the following formula, where \(g\) is the number of iterations. \[t(e_g)=\frac{(e_{g+w}-e_g)}{W}*10000\] On the rounded \(t(e_g)\) values, we find the first value of \(g\), \(g_{stabilization}\), when the rounded value of \(t(e_g) = 0\), and we stop if for 50 consecutive steps (i.e., \(g \in [g_{stabilization}..(g_{stabilization}+50)]\)), there is no change, \(t(e_g) = 0\) in this case, the stabilization time is the first moment where there was no change, namely \(g_{stabilization}\).
In our previous paper, Figure 13 shows the stabilization time for different network types (random, scale-free, and small-world). In this paper, we observed the stabilization time for two groups of agents (biased and unbiased). Overall, we observed that the stabilization time was never higher than 1000 iterations for any of the networks or any of the groups.
Since the model used in this paper is very similar (except that the language used is not binary anymore but multinomial or continuous), we expect similar results. However, due to the change in the type of language, we ran a few extra simulations to make sure these conclusions are transferable.
We observe what happens with time (from round 1 to round 4000) in a scale-free network containing 150 nodes. We chose to look specifically at scale-free networks because it was the type of network that requires the most time to stabilize (see our previous paper -Josserand et al. (2021)). We observe the evolution of different measures: inter-individual variation, the mean intra-individual variation and the std of intra-individual variation.
We look here at 10 different simulations :
In the following graph, each simulation is characterized by its color. The vertical plain lines shows the stabilization value based on the method presented above. The vertical black dashed line shows our choice for selecting the final value of the language.
Figure 1. Looking at the evolution of our different measures (top: inter-individual variability middle: mean of intra-individual variability bottom: std of intra-individual variability) with time. All agents speaks once in one interaction. We observe here 10 different simulations. Each simulation has a specific color. The plain vertical line shows the stabilization time for each replication (corresponding colors). The dashed black line shows our choice for selecting the final value of the language.
According to this data, our measures of variation have always stabilized after 1500 interations.
However, we decided here to study the language after agents have interacted during 3000 iterations. Indeed, it does not require much more time to compute, and we make sure that the language will indeed be stabilized in all our simulations.
We did exactly the same but using the language simulations where language is repsented as a continuous feature..
We observe the evolution of different measures: inter-individual variation, the mean intra-individual variation and the std of intra-individual variation.
We look here at 10 different simulations :
In the following graph, each simulation is characterized by its color.
Please note that in the first graph below (inter-individual variation), it seems that there is only one simulation (one color), but this is only the case because all simulations have almost the same values, so the points are superimposed.
Figure 2. Looking at the evolution of our different measures (top: inter-individual variability middle: mean of intra-individual variability bottom: std of intra-individual variability) with time. All agents speaks once in one interaction. We observe here 10 different simulations. Each simulation has a specific color. The plain vertical line shows the stabilization time for each replication (corresponding colors). The dashed black line shows our choice for selecting the final value of the language.
Popualtions with continuous language take way less time to stabilize. For simplicity reasons, we use the same threshold as for multinomial language.
Summary:
We record the final value of the language after 3000 iterations.
This part first presents the detailed results for each metric. However, you can refer to the Summary multinomial part, where you can see all the information in one plot.
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low pathlength set | 3.374239 | 1.96 | 0 | -0.3125999 | 2.798979 | 50 | Scalefree |
| High pathlength set | 4.987007 | 1.96 | 0 | -0.3057365 | 2.790429 | 50 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in pathlength, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 3. Looking at the variability inter- and intra-individual in sets varying in pathlength. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| pathlength | Inter | Without | 1.464 | 4 | 1.150 | 1.778 |
| pathlength | Inter | With (weak) | 1.522 | 4 | 1.205 | 1.838 |
| pathlength | Inter | With (strong) | 1.247 | 4 | 0.942 | 1.552 |
| pathlength | Mean Intra | Without | 0.070 | 1 | -0.209 | 0.349 |
| pathlength | Mean Intra | With (weak) | 0.308 | 2 | 0.588 | 0.027 |
| pathlength | Mean Intra | With (strong) | 0.023 | 1 | -0.256 | 0.302 |
| pathlength | Std Intra | Without | 0.296 | 2 | 0.016 | 0.577 |
| pathlength | Std Intra | With (weak) | 0.677 | 3 | 0.390 | 0.964 |
| pathlength | Std Intra | With (strong) | 0.505 | 3 | 0.221 | 0.788 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| pathlength | Inter | Without | 10.351 | 185.325 | 0.0000000 | 1.150 | 1.778 |
| pathlength | Inter | With (weak) | 10.760 | 173.081 | 0.0000000 | 1.205 | 1.838 |
| pathlength | Inter | With (strong) | 8.819 | 194.129 | 0.0000000 | 0.942 | 1.552 |
| pathlength | Mean Intra | Without | 0.493 | 183.826 | 0.6226403 | -0.209 | 0.349 |
| pathlength | Mean Intra | With (weak) | -2.176 | 183.850 | 0.0308327 | -0.588 | -0.027 |
| pathlength | Mean Intra | With (strong) | -0.163 | 177.904 | 0.8706430 | -0.302 | 0.256 |
| pathlength | Std Intra | Without | 2.095 | 194.168 | 0.0374853 | 0.016 | 0.577 |
| pathlength | Std Intra | With (weak) | 4.789 | 196.574 | 0.0000033 | 0.390 | 0.964 |
| pathlength | Std Intra | With (strong) | 3.568 | 192.189 | 0.0004535 | 0.221 | 0.788 |
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low assortativity set | 4.007469 | 1.96 | 0 | -0.4832008 | 2.747048 | 50 | Scalefree |
| High assortativity set | 4.016071 | 1.96 | 0 | -0.1650383 | 2.752900 | 50 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in assortativity, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 4. Looking at the variability inter- and intra-individual in sets varying in assortativity. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
We want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| assortativity | Inter | Without | 0.405 | 2 | 0.686 | 0.123 |
| assortativity | Inter | With (weak) | 0.392 | 2 | 0.673 | 0.110 |
| assortativity | Inter | With (strong) | 0.140 | 1 | -0.139 | 0.419 |
| assortativity | Mean Intra | Without | 0.467 | 2 | 0.184 | 0.750 |
| assortativity | Mean Intra | With (weak) | 0.083 | 1 | -0.196 | 0.362 |
| assortativity | Mean Intra | With (strong) | 0.023 | 1 | -0.256 | 0.302 |
| assortativity | Std Intra | Without | 0.234 | 2 | -0.046 | 0.514 |
| assortativity | Std Intra | With (weak) | 0.050 | 1 | -0.229 | 0.329 |
| assortativity | Std Intra | With (strong) | 0.034 | 1 | -0.245 | 0.313 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| assortativity | Inter | Without | -2.861 | 187.026 | 0.0047091 | -0.686 | -0.123 |
| assortativity | Inter | With (weak) | -2.771 | 192.766 | 0.0061313 | -0.673 | -0.110 |
| assortativity | Inter | With (strong) | -0.988 | 181.943 | 0.3244111 | -0.419 | 0.139 |
| assortativity | Mean Intra | Without | 3.303 | 174.885 | 0.0011619 | 0.184 | 0.750 |
| assortativity | Mean Intra | With (weak) | 0.587 | 196.558 | 0.5580783 | -0.196 | 0.362 |
| assortativity | Mean Intra | With (strong) | -0.165 | 194.361 | 0.8694981 | -0.302 | 0.256 |
| assortativity | Std Intra | Without | -1.655 | 192.451 | 0.0996226 | -0.514 | 0.046 |
| assortativity | Std Intra | With (weak) | 0.355 | 188.545 | 0.7227121 | -0.229 | 0.329 |
| assortativity | Std Intra | With (strong) | 0.242 | 175.069 | 0.8089962 | -0.245 | 0.313 |
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low exponent set | 4.101233 | 1.96 | 0 | -0.3545311 | 2.552332 | 50 | Scalefree |
| High exponent set | 4.093472 | 1.96 | 0 | -0.3459061 | 3.037600 | 50 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in exponent, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 5. Looking at the variability inter- and intra-individual in sets varying in exponent. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| exponent | Inter | Without | 0.313 | 2 | 0.593 | 0.032 |
| exponent | Inter | With (weak) | 0.330 | 2 | 0.611 | 0.049 |
| exponent | Inter | With (strong) | 0.152 | 1 | -0.128 | 0.431 |
| exponent | Mean Intra | Without | 0.081 | 1 | -0.198 | 0.360 |
| exponent | Mean Intra | With (weak) | 0.026 | 1 | -0.252 | 0.305 |
| exponent | Mean Intra | With (strong) | 0.076 | 1 | -0.203 | 0.355 |
| exponent | Std Intra | Without | 0.040 | 1 | -0.239 | 0.319 |
| exponent | Std Intra | With (weak) | 0.316 | 2 | 0.597 | 0.035 |
| exponent | Std Intra | With (strong) | 0.120 | 1 | -0.159 | 0.399 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| exponent | Inter | Without | -2.212 | 191.145 | 0.0281752 | -0.593 | -0.032 |
| exponent | Inter | With (weak) | -2.332 | 195.703 | 0.0206957 | -0.611 | -0.049 |
| exponent | Inter | With (strong) | -1.077 | 197.651 | 0.2827301 | -0.432 | 0.127 |
| exponent | Mean Intra | Without | 0.571 | 197.918 | 0.5688533 | -0.198 | 0.360 |
| exponent | Mean Intra | With (weak) | -0.181 | 197.846 | 0.8569108 | -0.304 | 0.253 |
| exponent | Mean Intra | With (strong) | 0.537 | 190.474 | 0.5920046 | -0.203 | 0.355 |
| exponent | Std Intra | Without | -0.285 | 196.407 | 0.7757957 | -0.319 | 0.239 |
| exponent | Std Intra | With (weak) | -2.235 | 196.847 | 0.0265320 | -0.597 | -0.035 |
| exponent | Std Intra | With (strong) | 0.846 | 197.731 | 0.3984539 | -0.159 | 0.399 |
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low size set | 4.490076 | 1.96 | 0 | -0.2974416 | 2.848549 | 50 | Scalefree |
| High size set | 4.499676 | 1.98 | 0 | -0.2908039 | 2.851606 | 100 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in size, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 6. Looking at the variability inter- and intra-individual in sets varying in size. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| size | Inter | Without | 0.894 | 4 | 0.601 | 1.186 |
| size | Inter | With (weak) | 0.492 | 2 | 0.209 | 0.775 |
| size | Inter | With (strong) | 0.304 | 2 | 0.024 | 0.585 |
| size | Mean Intra | Without | 0.112 | 1 | -0.167 | 0.391 |
| size | Mean Intra | With (weak) | 0.040 | 1 | -0.238 | 0.319 |
| size | Mean Intra | With (strong) | 0.104 | 1 | -0.175 | 0.383 |
| size | Std Intra | Without | 0.381 | 2 | 0.100 | 0.663 |
| size | Std Intra | With (weak) | 0.340 | 2 | 0.059 | 0.621 |
| size | Std Intra | With (strong) | 0.042 | 1 | -0.237 | 0.321 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| size | Inter | Without | 6.321 | 197.778 | 0.0000000 | 0.601 | 1.186 |
| size | Inter | With (weak) | 3.479 | 197.998 | 0.0006180 | 0.209 | 0.775 |
| size | Inter | With (strong) | 2.153 | 192.077 | 0.0325857 | 0.024 | 0.585 |
| size | Mean Intra | Without | -0.792 | 195.438 | 0.4294977 | -0.391 | 0.167 |
| size | Mean Intra | With (weak) | 0.286 | 188.377 | 0.7749995 | -0.238 | 0.319 |
| size | Mean Intra | With (strong) | 0.734 | 190.333 | 0.4639971 | -0.175 | 0.383 |
| size | Std Intra | Without | 2.695 | 188.560 | 0.0076647 | 0.100 | 0.663 |
| size | Std Intra | With (weak) | 2.406 | 192.068 | 0.0170732 | 0.059 | 0.621 |
| size | Std Intra | With (strong) | 0.298 | 195.277 | 0.7660905 | -0.237 | 0.321 |
These sets of networks were obtained using
small-world networks, and varying the parameters
rewire.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low clustering set | 1.677852 | 48 | 0.3189198 | -0.0145174 | NA | 150 | Small-world |
| High clustering set | 1.682844 | 48 | 0.5826920 | -0.0065716 | NA | 150 | Small-world |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in clustering, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 7. Looking at the variability inter- and intra-individual in sets varying in clustering. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Please note that the x scale here is different due to
differences in the network type.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| clustering | Inter | Without | 4.163 | 4 | 3.667 | 4.660 |
| clustering | Inter | With (weak) | 4.160 | 4 | 3.664 | 4.656 |
| clustering | Inter | With (strong) | 3.420 | 4 | 2.983 | 3.858 |
| clustering | Mean Intra | Without | 0.050 | 1 | -0.229 | 0.329 |
| clustering | Mean Intra | With (weak) | 0.087 | 1 | -0.192 | 0.366 |
| clustering | Mean Intra | With (strong) | 0.296 | 2 | 0.576 | 0.015 |
| clustering | Std Intra | Without | 2.017 | 4 | 1.674 | 2.359 |
| clustering | Std Intra | With (weak) | 2.220 | 4 | 1.866 | 2.575 |
| clustering | Std Intra | With (strong) | 2.070 | 4 | 1.724 | 2.415 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| clustering | Inter | Without | 29.440 | 99.004 | 0.0000000 | 3.667 | 4.660 |
| clustering | Inter | With (weak) | 29.416 | 99.007 | 0.0000000 | 3.664 | 4.656 |
| clustering | Inter | With (strong) | 24.184 | 99.015 | 0.0000000 | 2.983 | 3.858 |
| clustering | Mean Intra | Without | 0.356 | 193.893 | 0.7223852 | -0.229 | 0.329 |
| clustering | Mean Intra | With (weak) | 0.618 | 197.975 | 0.5369757 | -0.192 | 0.366 |
| clustering | Mean Intra | With (strong) | -2.092 | 194.149 | 0.0377403 | -0.576 | -0.015 |
| clustering | Std Intra | Without | 14.259 | 99.958 | 0.0000000 | 1.674 | 2.359 |
| clustering | Std Intra | With (weak) | 15.701 | 99.653 | 0.0000000 | 1.866 | 2.575 |
| clustering | Std Intra | With (strong) | 14.635 | 99.254 | 0.0000000 | 1.724 | 2.415 |
These sets of networks were obtained using random networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low node degree set | 1.785524 | 11.97893 | 0.2450417 | 0.0138245 | NA | 50 | Random |
| High node degree set | 1.777197 | 12.69280 | 0.2531758 | 0.0249452 | NA | 50 | Random |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in node degree, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 8. Looking at the variability inter- and intra-individual in sets varying in node degree. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Please note that the x scale here is different due to
differences in the network type.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| node_degree | Inter | Without | 0.708 | 3 | 0.995 | 0.420 |
| node_degree | Inter | With (weak) | 0.725 | 3 | 1.013 | 0.437 |
| node_degree | Inter | With (strong) | 0.648 | 3 | 0.935 | 0.362 |
| node_degree | Mean Intra | Without | 0.140 | 1 | -0.139 | 0.420 |
| node_degree | Mean Intra | With (weak) | 0.268 | 2 | -0.012 | 0.548 |
| node_degree | Mean Intra | With (strong) | 0.311 | 2 | 0.592 | 0.031 |
| node_degree | Std Intra | Without | 0.057 | 1 | -0.222 | 0.336 |
| node_degree | Std Intra | With (weak) | 0.138 | 1 | -0.141 | 0.418 |
| node_degree | Std Intra | With (strong) | 0.298 | 2 | 0.578 | 0.017 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| node_degree | Inter | Without | -5.006 | 191.822 | 0.0000013 | -0.995 | -0.420 |
| node_degree | Inter | With (weak) | -5.125 | 197.972 | 0.0000007 | -1.013 | -0.437 |
| node_degree | Inter | With (strong) | -4.585 | 195.159 | 0.0000081 | -0.935 | -0.362 |
| node_degree | Mean Intra | Without | -0.987 | 197.253 | 0.3248058 | -0.419 | 0.140 |
| node_degree | Mean Intra | With (weak) | -1.894 | 191.699 | 0.0597234 | -0.548 | 0.012 |
| node_degree | Mean Intra | With (strong) | -2.202 | 197.439 | 0.0288564 | -0.592 | -0.031 |
| node_degree | Std Intra | Without | 0.406 | 196.701 | 0.6854013 | -0.222 | 0.336 |
| node_degree | Std Intra | With (weak) | -0.974 | 197.709 | 0.3313699 | -0.417 | 0.142 |
| node_degree | Std Intra | With (strong) | -2.105 | 191.655 | 0.0365533 | -0.578 | -0.017 |
These sets of networks were obtained using random networks. Pathlength was already explored using scale-free networks, and controlling for many metrics Here, we just want to observe the effect of pathlength in opposite to the effect of mean degree, in random network.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low pathlength set | 4.314229 | 2.467467 | 0.0285249 | -0.1041842 | NA | 50 | Random |
| High pathlength set | 4.314229 | 2.467467 | 0.0285249 | -0.1041842 | NA | 50 | Random |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in node degree, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 9. Looking at the variability inter- and intra-individual in sets varying in pathlength (using random networks, and not scale-free networks). The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Please note that the x scale here is different due to
differences in the network type.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| pathlength_ran | Inter | Without | 0.427 | 2 | 0.145 | 0.709 |
| pathlength_ran | Inter | With (weak) | 0.727 | 3 | 0.439 | 1.014 |
| pathlength_ran | Inter | With (strong) | 0.512 | 3 | 0.229 | 0.796 |
| pathlength_ran | Mean Intra | Without | 0.001 | 1 | -0.278 | 0.279 |
| pathlength_ran | Mean Intra | With (weak) | 0.000 | 1 | 0.279 | 0.279 |
| pathlength_ran | Mean Intra | With (strong) | 0.023 | 1 | -0.256 | 0.302 |
| pathlength_ran | Std Intra | Without | 0.134 | 1 | -0.145 | 0.413 |
| pathlength_ran | Std Intra | With (weak) | 0.380 | 2 | 0.099 | 0.661 |
| pathlength_ran | Std Intra | With (strong) | 0.265 | 2 | -0.015 | 0.545 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| pathlength_ran | Inter | Without | 3.022 | 196.497 | 0.0028414 | 0.145 | 0.709 |
| pathlength_ran | Inter | With (weak) | 5.137 | 183.653 | 0.0000007 | 0.439 | 1.014 |
| pathlength_ran | Inter | With (strong) | 3.623 | 185.944 | 0.0003750 | 0.229 | 0.796 |
| pathlength_ran | Mean Intra | Without | 0.004 | 193.154 | 0.9969390 | -0.278 | 0.279 |
| pathlength_ran | Mean Intra | With (weak) | 0.001 | 192.635 | 0.9990379 | -0.279 | 0.279 |
| pathlength_ran | Mean Intra | With (strong) | 0.161 | 196.648 | 0.8721028 | -0.256 | 0.302 |
| pathlength_ran | Std Intra | Without | 0.949 | 198.000 | 0.3438721 | -0.145 | 0.413 |
| pathlength_ran | Std Intra | With (weak) | 2.686 | 196.273 | 0.0078432 | 0.099 | 0.661 |
| pathlength_ran | Std Intra | With (strong) | 1.872 | 196.351 | 0.0627478 | -0.015 | 0.545 |
Plot with full data:
You can find below the plot used in the main paper, which shows exactly the same information except that we do not present the results for initial language exposure = With (Strong). Also, we renamed the initial language exposure with type of language scenario (without initial language exposure -> Language emergence and with (weak) initial language exposure -> Language change). See Note on Initial language exposure terminology for more information.
This part shows exactly the same type of results, except that here language is continuous and not multinomial. We present the detailed results for each metric. However, you can refer to the Summary continuous part, where you can see all the information in one plot. Please also note that we do not contrast language emergence and language change here.
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low pathlength set | 3.392992 | 2 | 0 | -0.3101820 | 2.783516 | 50 | Scalefree |
| High pathlength set | 4.982433 | 2 | 0 | -0.3068284 | 2.770500 | 50 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in pathlength, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 10. Looking at the variability inter- and intra-individual in sets varying in pathlength in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| pathlength | Inter | None | 6.427 | large | 7.119 | 5.734 |
| pathlength | Mean Intra | None | 0.676 | medium | 0.389 | 0.963 |
| pathlength | Std Intra | None | 3.292 | large | 2.864 | 3.720 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| pathlength | Inter | None | -45.444 | 171.791 | 0.0e+00 | 7.119 | 5.734 |
| pathlength | Mean Intra | None | 4.781 | 187.838 | 3.5e-06 | 0.389 | 0.963 |
| pathlength | Std Intra | None | 23.279 | 189.202 | 0.0e+00 | 2.864 | 3.720 |
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low assortativity set | 3.998033 | 2 | 0 | -0.4584237 | 2.781426 | 50 | Scalefree |
| High assortativity set | 4.008808 | 2 | 0 | -0.1753525 | 2.785640 | 50 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in assortativity, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 11. Looking at the variability inter- and intra-individual in sets varying in assortativity in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
COhen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| assortativity | Inter | None | 5.139 | large | 5.717 | 4.561 |
| assortativity | Mean Intra | None | 0.232 | small | 0.512 | 0.048 |
| assortativity | Std Intra | None | 2.174 | large | 1.822 | 2.526 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| assortativity | Inter | None | -36.338 | 188.901 | 0.0000000 | 0.000 | 0 |
| assortativity | Mean Intra | None | -1.643 | 190.737 | 0.1021278 | 0.003 | 0 |
| assortativity | Std Intra | None | 15.373 | 192.193 | 0.0000000 | 0.000 | 0 |
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low exponent set | 4.003612 | 2 | 0 | -0.2662336 | 2.610069 | 50 | Scalefree |
| High exponent set | 3.988955 | 2 | 0 | -0.2615309 | 3.067970 | 50 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in exponent, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 12. Looking at the variability inter- and intra-individual in sets varying in exponent in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| exponent | Inter | None | 2.357 | large | 2.720 | 1.994 |
| exponent | Mean Intra | None | 0.033 | negligible | 0.312 | 0.246 |
| exponent | Std Intra | None | 3.359 | large | 3.792 | 2.926 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| exponent | Inter | None | -16.667 | 190.982 | 0.0000000 | 0.000 | 0.000 |
| exponent | Mean Intra | None | -0.233 | 197.326 | 0.8157111 | 0.002 | 0.001 |
| exponent | Std Intra | None | -23.753 | 176.871 | 0.0000000 | 0.000 | 0.000 |
These sets of networks were obtained using scale-free networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low size set | 4.428812 | 2 | 0 | -0.2983585 | 2.845419 | 50 | Scalefree |
| High size set | 4.439875 | 2 | 0 | -0.2902309 | 2.854948 | 100 | Scalefree |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in size, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 13. Looking at the variability inter- and intra-individual in sets varying in size in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| size | Inter | None | 4.608 | large | 4.075 | 5.141 |
| size | Mean Intra | None | 0.213 | small | 0.067 | 0.493 |
| size | Std Intra | None | 2.871 | large | 3.269 | 2.474 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| size | Inter | None | 32.586 | 177.690 | 0.0000000 | 0 | 0.000 |
| size | Mean Intra | None | 1.505 | 187.882 | 0.1339995 | 0 | 0.003 |
| size | Std Intra | None | -20.304 | 189.611 | 0.0000000 | 0 | 0.000 |
These sets of networks were obtained using
small-world networks, and varying the parameters
rewire.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low clustering set | 1.677852 | 48 | 0.3191316 | -0.0151775 | NA | 150 | Small-world |
| High clustering set | 1.682726 | 48 | 0.5820881 | -0.0041973 | NA | 150 | Small-world |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in clustering, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 14. Looking at the variability inter- and intra-individual in sets varying in clustering coefficient in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Please note that the x scale here is different due to
differences in the network type.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| clustering | Inter | None | 9.538 | large | 10.519 | 8.557 |
| clustering | Mean Intra | None | 2.721 | large | 2.334 | 3.108 |
| clustering | Std Intra | None | 11.598 | large | 12.775 | 10.421 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| clustering | Inter | None | -67.443 | 109.604 | 0 | 0.000 | 0.000 |
| clustering | Mean Intra | None | 19.242 | 99.004 | 0 | 0.001 | 0.001 |
| clustering | Std Intra | None | -82.009 | 141.684 | 0 | 0.000 | 0.000 |
These sets of networks were obtained using random networks.
We first observe the values of the two sets:
| pathlength | neighbors | clustering | assortativity | exponent | size | network_type | |
|---|---|---|---|---|---|---|---|
| Low node degree set | 1.785351 | 12.0072 | 0.244033 | 0.0116981 | NA | 50 | Random |
| High node degree set | 1.785351 | 12.0072 | 0.244033 | 0.0116981 | NA | 50 | Random |
This should be interpreted in the following way:
Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in node degree, as the other metrics are kept (almost) constant.
We look at the differences between the two sets:
Figure 15. Looking at the variability inter- and intra-individual in sets varying in node degree in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.
Please note that the x scale here is different due to
differences in the network type.
Here, we want to understand whether the effect size under these differences is high or low.
We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:
Cohen’s D:
| Measure | TypeVariation | TypeLangage | CohenD | Magnitude | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|
| node_degree | Inter | None | 2.183 | large | 2.535 | 1.830 |
| node_degree | Mean Intra | None | 0.038 | negligible | 0.240 | 0.317 |
| node_degree | Std Intra | None | 0.981 | large | 0.686 | 1.277 |
T-Test:
| Measure | TypeVariation | TypeLangage | T_value | DF | PValue | CI_Inf | CI_Sup |
|---|---|---|---|---|---|---|---|
| node_degree | Inter | None | -15.433 | 159.110 | 0.0000000 | 0 | 0 |
| node_degree | Mean Intra | None | 0.272 | 196.039 | 0.7860134 | 0 | 0 |
| node_degree | Std Intra | None | 6.939 | 116.802 | 0.0000000 | 0 | 0 |
In this dataset, we generated thousands of simulations using random, small-world and scalefree networks. Using this dataset, we aim to:
While in dataset 1 we use specific sets of networks (which can be seen as “unnatural”), in Dataset 2, we observe “natural” networks (namely, we did not choose to exclude any of the networks generated). Please note that here we only analyzed a multinomial language (contrary to Dataset 1, we do not study data from networks with continuous language). You can refer to Parameters to know more about the parameters which were investigated, but please note that a summary (and data distribution) of the results is also presented below.
Please note that exponent is a metric considered only in scale-free networks. Additionnally, node degree and clustering coefficient are considered only in small-world and random network, since they do not vary in scale-free networks. We look at the data for each network type.
Random networks
| size | clustering | node_degree | pathlength | assortativity | init_lang_exp | connection_prob | inter_var | mean_intra_var | std_intra_var | |
|---|---|---|---|---|---|---|---|---|---|---|
| 50 :300 | Min. :0.0000 | Min. : 2.20 | Min. :1.043 | Min. :-0.241523 | With (strong):298 | Min. :0.0500 | Min. :6.300e-08 | Min. :0.7455 | Min. :3.542e-05 | |
| 150:295 | 1st Qu.:0.2506 | 1st Qu.: 22.71 | 1st Qu.:1.250 | 1st Qu.:-0.031982 | With (weak) :298 | 1st Qu.:0.2500 | 1st Qu.:1.264e-06 | 1st Qu.:1.3053 | 1st Qu.:5.088e-04 | |
| 300:300 | Median :0.5294 | Median : 51.09 | Median :1.469 | Median :-0.013001 | Without :299 | Median :0.5500 | Median :4.474e-06 | Median :1.9657 | Median :1.192e-03 | |
| NA | Mean :0.5010 | Mean : 83.12 | Mean :1.620 | Mean :-0.021829 | NA | Mean :0.5015 | Mean :3.374e-04 | Mean :1.8145 | Mean :3.937e-03 | |
| NA | 3rd Qu.:0.7495 | 3rd Qu.:126.51 | 3rd Qu.:1.751 | 3rd Qu.:-0.005664 | NA | 3rd Qu.:0.7500 | 3rd Qu.:1.702e-05 | 3rd Qu.:2.2386 | 3rd Qu.:2.611e-03 | |
| NA | Max. :0.9568 | Max. :284.75 | Max. :4.985 | Max. : 0.096417 | NA | Max. :0.9500 | Max. :1.957e-02 | Max. :2.3002 | Max. :1.242e-01 |
Quick note: the number of random network is not exactly equal to 900 because we removed a few networks where there were isolated components.
In random networks, the only structural parameters we manipulated were the connection probability (from 0.05 to 0.95 in steps of 0.05) and the size (50, 150, and 300).
Figure 16. Distribution of the data in random networks.
Scale-free networks
| size | exponent | pathlength | assortativity | init_lang_exp | inter_var | mean_intra_var | std_intra_var | |
|---|---|---|---|---|---|---|---|---|
| 50 :300 | Min. :2.416 | Min. :2.820 | Min. :-0.5590 | With (strong):300 | Min. :0.002436 | Min. :0.9714 | Min. :0.01744 | |
| 150:300 | 1st Qu.:2.887 | 1st Qu.:4.218 | 1st Qu.:-0.2991 | With (weak) :300 | 1st Qu.:0.008562 | 1st Qu.:1.2778 | 1st Qu.:0.04448 | |
| 300:300 | Median :3.026 | Median :4.980 | Median :-0.2310 | Without :300 | Median :0.020193 | Median :1.9350 | Median :0.08100 | |
| NA | Mean :2.998 | Mean :4.941 | Mean :-0.2495 | NA | Mean :0.020313 | Mean :1.7960 | Mean :0.07720 | |
| NA | 3rd Qu.:3.122 | 3rd Qu.:5.645 | 3rd Qu.:-0.1879 | NA | 3rd Qu.:0.029502 | 3rd Qu.:2.2091 | 3rd Qu.:0.10310 | |
| NA | Max. :3.367 | Max. :7.285 | Max. :-0.0873 | NA | Max. :0.047241 | Max. :2.2538 | Max. :0.17089 |
In scale-free networks, the only structural parameter we manipulated was the size (50, 150, and 300).
Figure 17. Distribution of the data in scale-free networks.
Small-world networks
| size | clustering | node_degree | pathlength | assortativity | init_lang_exp | rewire | inter_var | mean_intra_var | std_intra_var | |
|---|---|---|---|---|---|---|---|---|---|---|
| 50 :300 | Min. :0.001556 | Min. : 4 | Min. :1.020 | Min. :-0.29315 | With (strong):300 | Min. :0.1 | Min. :2.030e-07 | Min. :0.8495 | Min. :0.0001095 | |
| 150:300 | 1st Qu.:0.100506 | 1st Qu.: 7 | 1st Qu.:1.829 | 1st Qu.:-0.06832 | With (weak) :300 | 1st Qu.:0.3 | 1st Qu.:4.184e-05 | 1st Qu.:1.2909 | 1st Qu.:0.0028076 | |
| 300:300 | Median :0.232127 | Median :12 | Median :2.364 | Median :-0.03556 | Without :300 | Median :0.5 | Median :3.634e-04 | Median :1.9660 | Median :0.0087059 | |
| NA | Mean :0.296572 | Mean :19 | Mean :2.635 | Mean :-0.04476 | NA | Mean :0.5 | Mean :2.291e-03 | Mean :1.8188 | Mean :0.0184484 | |
| NA | 3rd Qu.:0.399529 | 3rd Qu.:24 | 3rd Qu.:3.071 | 3rd Qu.:-0.01115 | NA | 3rd Qu.:0.7 | 3rd Qu.:2.709e-03 | 3rd Qu.:2.2501 | 3rd Qu.:0.0275223 | |
| NA | Max. :0.980213 | Max. :48 | Max. :7.277 | Max. : 0.11942 | NA | Max. :0.9 | Max. :3.078e-02 | Max. :2.2993 | Max. :0.1244774 |
In small-world networks, the only structural parameters we manipulated were the rewiring probability (0.1, 0.3, 0.5, 0.7, 0.9), the node degree (4, 8, 16, 48 neighbors), and the size (50, 150, and 300).
For simplicity reasons, we do not plot here the node degree as a facet.Figure 18. Distribution of the data in small-world networks.
The data is not normally distributed. So we should use Spearman’s correlations instead of Pearson’s correlations. Since it does not cost much to compute both, you can find below both of them :)
Please note that I didn’t manage to put them in grid.arrange all together.
Pearson’s correlations:
Figure 19. Pearsons correlations.
Figure 19. Pearsons correlations.
Figure 19. Pearsons correlations.
Spearman’s correlations:
Figure 20. Spearmans correlations.
Figure 20. Spearmans correlations.
Figure 20. Spearmans correlations.
We found high correlations between some variables. In order to understand the direction of correlation, we graphically print the relationships between the metrics in all types of networks. Size will always be printed in a color layer.
Random networks:
Figure 21. Graphical visualization of the relationship between the metrics using a range of random networks.
Please note that we ran several simulation with random network using different connection probability and size of the network, which correspond to the different bumps in the visualization.
Small-world networks:
Figure 22. Graphical visualization of the relationship between the metrics using a range of small-world networks.
Please note that we ran several simulation with small-world network using different rewiring probability, node degree, and size of the network, which correspond to the different bumps in the visualization.
Scale-free networks:
Figure 23. Graphical visualization of the relationship between the metrics using a range of scale-free networks.
We also look at the PCA between these metrics in all types of networks.
Random networks:
Figure 24. Histogram of the eigenvalues of the PCA between the metrics using a range of random networks.
Figure 25. Contribution of the different metrics in the PCA using a range of random networks.
Small-world networks:
Figure 27. Histogram of the eigenvalues of the PCA between the metrics using a range of small-world networks.
Figure 28. Contribution of the different metrics in the PCA using a range of small-world networks.
Scale-free networks:
Figure 30. Histogram of the eigenvalues of the PCA between the metrics using a range of scale-free networks.
Figure 31. Contribution of the different metrics in the PCA using a range of scale-free networks.
We predict variation (inter, mean intra, and std intra) from a collection of potential predictors using three machine learning techniques:
random forests (RF): as implemented by
randomForest() in package randomForest. This
function implements Breiman’s random forest algorithm (based on Breiman
and Cutler’s original Fortran code); see here
for more information. RF have been shown to be particularly relevant
when predictors are inter-correlated (see Levshina, 2021, in the Chapter
25 “Conditional Inference Trees and Random Forests”). We extract the
predictor importance using two indexes: the accuracy-based
index and the gini-based index. The accuracy-based index
measures the reduction in model accuracy if a particular predictor is
removed or permuted. On the other hand, the Gini Index is a measure of
impurity or purity in a decision tree. In the context of random forests,
namely, it measures how often a randomly chosen element from the dataset
would be incorrectly labeled if it was randomly labeled according to the
distribution of labels in the node.
conditional random forests (CF): as implemented
by cforest() in package partykit. The number
of trees is 200, and the number of permutations is 10. We measured the
predictor importance using the unconditional index. The
unconditional index for a given predictor measures how well that
predictor alone, in isolation, can help to improve the predictive
accuracy of the model.
support vector machines (SVM): as implemented by
fit(...,model="svm") in the rminer package;
see here). We
used a regression task (since the output is continuous). This function
uses ksvm from kernlab package; more
information on this function can be found here.
We measured the predictor importance based on sensitivity
analysis. The sensitivity method for predictor importance in SVM
involves perturbing individual predictors to observe the resulting
changes in the model’s output, quantifying the impact, and ranking
predictors based on their sensitivity scores to identify the most
influential features in the SVM decision-making process.
For each method, we extract some success measures, here the
R² and RMSE using postResample function of
rminer package. Note that we z-scored all the
predictors.
Success in language emergence context:
Inter-individual variation:
Mean Intra-individual variation:
Std Intra-individual variation:
Let’s plot this data:
Figure 33. Success (using R² and RMSE indexes) in random networks in language emergence scenarios for each type of method.
Predictor importance in language emergence context:
Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!
Figure 34. Predictor importance in random networks in language emergence scenarios for each type of method.
Success in language change context:
Inter-individual variation:
Mean Intra-individual variation:
Std Intra-individual variation:
Figure 35. Success (using R² and RMSE indexes) in random networks in language change scenario for each type of method.
Predictor importance in language change context:
Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!
Figure 36. Predictor importance in random networks in language change scenario for each type of method.
Success in language emergence context:
Inter-individual variation:
Mean Intra-individual variation:
Std Intra-individual variation:
Figure 37. Success (using R² and RMSE indexes) in small-world networks in language emergence scenario for each type of method.
Predictor importance in language emergence context:
Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!
Figure 38. Predictor importance in small-world networks in language emergence scenario for each type of method.
Success in language change context:
Inter-individual variation:
Mean Intra-individual variation:
Std Intra-individual variation:
Figure 39. Success in small-world networks in language change scenario for each type of method.
Predictor importance in language change context:
Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!
Figure 40. Predictor importance in small-world networks in language change scenario for each type of method.
Success in language emergence context:
Inter-individual variation:
Mean Intra-individual variation:
Std Intra-individual variation:
Figure 41. Success (using R² and RMSE indexes) in scale-free networks in language emergence scenario for each type of method.
Predictor importance in language emergence context:
And now, let’s visualize the importance of the predictors. Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!
Figure 42. Predictor importance in scale-free networks in language emergence scenario for each type of method.
Success in language change context:
Inter-individual variation:
Mean Intra-individual variation:
Std Intra-individual variation:
Figure 43. Success (using R² and RMSE indexes) in scale-free networks in language change scenario for each type of method.
Predictor importance in language change context:
Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!
Figure 44. Predictor importance in scale-free networks in language change scenario for each type of method.
Success:
In language emergence scenario:
| Inter | Mintra | Sintra | |
|---|---|---|---|
| Random | 91.48819 | 45.324598 | 76.260186 |
| Small-World | 90.20976 | 58.875546 | 81.583876 |
| Scale-free | 37.50218 | 1.723138 | 4.176135 |
In language change scenario:
| Inter | Mintra | Sintra | |
|---|---|---|---|
| Random | 95.86643 | 2.4651129 | 90.334545 |
| Small-World | 88.03344 | 4.6870283 | 82.062926 |
| Scale-free | 36.76559 | 0.0151357 | 7.401045 |
The R² values are particularly low for intra-individual variation: 35.31% on average for all types of networks in language emergence and 2.39% for language change.
The R² values are also particularly low for scale-free variation: for both language emergence and language change, it predicts 37.5% for inter-individual variation and 4.18% for standard-deviation of intra-individual variation.
The rest is well-predicted: The mean R² of inter-individual variation for language emergence and change in random and small-world network is of 91.49%, while the standard-deviation of the intra-individual variation is of 76.26%.
Predictor importance:
Plot for the paper:
Figure 45. Mean normalized predictor importance (mean of the four index including the accuracy-based and gini indexes from random forests, the index from conditional forests and the index from SVM) in random et small-world networks. We look at inter-individual variation (left panel) and standard-deviation of intra-individual variation (right panel) in language change scenario (upper panel) and language emergence scenario (lower panel). Note that we selected only the models with high success of prediction.
Plot with all data:
Figure 46. Mean normalized predictor importance (mean of the four index including the accuracy-based and gini indexes from random forests, the index from conditional forests and the index from SVM) in random, small-world et scale-fre networks. We look at inter-individual variation (left panel), mean intra-individual variation (middle panel) and standard-deviation of intra-individual variation (right panel) in language change scenario (upper panel) and language emergence scenario (lower panel)
These results only concerns the machine learning techniques. They are corrobated by the use of linear models (see part below, or go directly to Summary Modeling part for a summary).
Then, we wish to observe the effect size of our different metrics
using modeling. In our models, we want to investigate the relative
effect of the different predictors (here, our metrics
normalized) on variation (our different measures for inter- and
intra- individual variability). Our measure of variation are all
continuous, so we use a linear model with function
lm (stats package, version 3.6.2, see here
for more information).
To do so, we create a model including all metrics as predictors. Then, using this model, we first look at the VIF score (Variance Inflation Factor). In statistics, the VIF score shows the extent of multicollinearity in a regression analysis. It is “the ratio (quotient) of the variance of estimating some parameter in a model that includes multiple other terms (parameters) by the variance of a model constructed using only one terms.”.
The extent to which a VIF score is “too high” is still debated. In general, it is suggested that a VIF score greater than 5 reflects highly correlated predictors (source: here or here), but please note that there is no absolute “truth” for what is a high VIF score.
In scale-free networks:
In language emergence:
| metrics | vif_score |
|---|---|
| exponent_z | 2.094996 |
| assortativity_z | 2.176258 |
| pathlength_z | 3.122374 |
| size_z | 4.559353 |
In language change:
| metrics | vif_score |
|---|---|
| exponent_z | 2.232382 |
| assortativity_z | 2.052236 |
| pathlength_z | 2.936720 |
| size_z | 5.012959 |
We observe that no metrics have a VIF too high (above 5). Thus, we will keep all metrics in the model of scale-free networks.
In small-world networks:
In language emergence:
| metrics | vif_score |
|---|---|
| clustering_z | 2.212031 |
| node_degree_z | 3.581759 |
| size_z | 2.056694 |
| assortativity_z | 1.254807 |
| pathlength_z | 2.819969 |
In language change:
| metrics | vif_score |
|---|---|
| clustering_z | 2.256314 |
| node_degree_z | 3.630869 |
| size_z | 2.105092 |
| assortativity_z | 1.253142 |
| pathlength_z | 3.021626 |
We observe that no metrics have a VIF too high (above 5). Thus, we will keep all metrics in the model of small-world networks.
In random networks:
In language emergence:
| metrics | vif_score |
|---|---|
| node_degree_z | 7.368530 |
| size_z | 4.821028 |
| clustering_z | 7.776622 |
| pathlength_z | 3.560723 |
| assortativity_z | 1.380199 |
In language change:
| metrics | vif_score |
|---|---|
| node_degree_z | 7.450035 |
| size_z | 5.433643 |
| clustering_z | 7.624188 |
| pathlength_z | 3.393699 |
| assortativity_z | 1.581143 |
We observe that the following metrics have a VIF too high (above 5): clustering and node degree. However, the VIF score is not that much higher than 5 (7 for both metrics). To check which of these two metrics explains the most variation, we create models using only the two metrics (for both inter- and intra- individual variation):
Call:
lm(formula = inter_var * 100 ~ clustering_z + node_degree_z,
data = subdf_mod_ran_em)
Residuals:
Min 1Q Median 3Q Max
-0.17264 -0.10718 -0.03745 0.02481 1.77137
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.051867 0.014481 3.582 0.000399 ***
clustering_z -0.081600 0.018601 -4.387 1.6e-05 ***
node_degree_z -0.003229 0.018555 -0.174 0.861982
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2504 on 296 degrees of freedom
Multiple R-squared: 0.1006, Adjusted R-squared: 0.0945
F-statistic: 16.55 on 2 and 296 DF, p-value: 1.536e-07
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + node_degree_z,
data = subdf_mod_ran_em)
Residuals:
Min 1Q Median 3Q Max
-19.1812 -1.3586 0.2887 1.8521 8.1046
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 225.4401 0.1694 1330.79 <2e-16 ***
clustering_z -2.7810 0.2176 -12.78 <2e-16 ***
node_degree_z 3.0115 0.2171 13.87 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.929 on 296 degrees of freedom
Multiple R-squared: 0.4268, Adjusted R-squared: 0.4229
F-statistic: 110.2 on 2 and 296 DF, p-value: < 2.2e-16
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + node_degree_z,
data = subdf_mod_ran_em)
Residuals:
Min 1Q Median 3Q Max
-0.4323 -0.2181 -0.0823 0.0686 6.0083
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.17475 0.03273 5.339 1.86e-07 ***
clustering_z -0.19301 0.04204 -4.591 6.54e-06 ***
node_degree_z -0.03476 0.04194 -0.829 0.408
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.566 on 296 degrees of freedom
Multiple R-squared: 0.1278, Adjusted R-squared: 0.1219
F-statistic: 21.69 on 2 and 296 DF, p-value: 1.623e-09
We also did create models using only one predictor and looking at the amount of variation (R-squared) it explains, but it yield to the same conclusions: clearly, it seems that the clustering coefficient explains more variation than the node degree. Since the VIF is not that much higher than 5 and since this cutoff is arbitrary, we will first use a model containing all predictors, to ensure coherence with scale-free and small-world networks. At the end of the analysis, we will also remove node degree from the predictors, to make sure that this is does not affect our conclusions.
Summary
Here, we observe the diagnostic plots for all type of networks. Since the diagnostic plots for language change are extremely similar with the diagnostic plots for language emergence scenario, we present here the plots only for language emergence.
Random Networks:
In language emergence scenario:
Figure 47. Diagnostic plots when predicting inter-individual variation (random networks, language emergence).
Figure 48. Diagnostic plots when predicting mean intra-individual variation (random networks, language emergence)
Figure 49. Diagnostic plots when predicting standard-deviation of intra-individual variation (random networks, language emergence)
The model’s assumptions are not being met. Below, we will attempt to address this issue. Given the analogous patterns observed in both language emergence and language change scenarios, we will focus on correcting the model assumptions specifically in the context of language emergence and inter-individual variation to assess its impact.
It seems that linearity is not respected. Thus, we also check whether a quadratic model of these variables would fit better.
Call:
lm(formula = inter_var ~ size_z + clustering_z + I(clustering_z^2) +
node_degree_z + I(node_degree_z^2) + pathlength_z + I(pathlength_z^2) +
assortativity_z + I(assortativity_z^2), data = subdf_mod_ran_ch)
Residuals:
Min 1Q Median 3Q Max
-9.405e-04 -1.363e-05 3.360e-06 2.629e-05 1.777e-03
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.858e-04 3.085e-05 -9.264 < 2e-16 ***
size_z -5.751e-05 3.422e-05 -1.680 0.09397 .
clustering_z -8.792e-04 7.195e-05 -12.220 < 2e-16 ***
I(clustering_z^2) -1.685e-04 2.040e-05 -8.259 5.39e-15 ***
node_degree_z 1.362e-04 5.680e-05 2.397 0.01715 *
I(node_degree_z^2) -3.356e-05 1.701e-05 -1.974 0.04939 *
pathlength_z -1.355e-03 9.933e-05 -13.645 < 2e-16 ***
I(pathlength_z^2) 8.092e-04 1.882e-05 42.992 < 2e-16 ***
assortativity_z -5.673e-05 1.944e-05 -2.919 0.00379 **
I(assortativity_z^2) 4.255e-05 6.870e-06 6.194 2.02e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0001975 on 288 degrees of freedom
Multiple R-squared: 0.9882, Adjusted R-squared: 0.9879
F-statistic: 2684 on 9 and 288 DF, p-value: < 2.2e-16
It does not change much the results, and linearity is still not respected.
Shapiro-Wilk normality test
data: resid(model)
W = 0.45829, p-value < 2.2e-16
Significantly different from a normal distribution.
Non-constant Variance Score Test
Variance formula: ~ fitted.values
Chisquare = 2686.694, Df = 1, p = < 2.22e-16
The data has clearly heteroscedasticity. We log the output to decrease heteroscedasticity.
Non-constant Variance Score Test
Variance formula: ~ fitted.values
Chisquare = 2.681535, Df = 1, p = 0.10152
The data still has heteroscedasticity, but less.
scale-free Networks:
In language emergence scenario:
Figure 50. Diagnostic plots when predicting **inter*-individual variation (scale-free networks, language emergence)
Figure 51. Diagnostic plots when predicting mean intra-individual variation (scale-free networks, language emergence)
Figure 52. Diagnostic plots when predicting standard-deviation of intra-individual variation (scale-free networks, language emergence)
Small-world Networks:
In language emergence scenario:
Figure 53. Diagnostic plots when predicting inter-individual variation (small-world networks, language emergence)
Figure 54. Diagnostic plots when predicting mean intra-individual variation (small-world networks, language emergence)
Figure 55. Diagnostic plots when predicting standard-deviation of intra-individual variation (small-world networks, language emergence)
First, we create a linear model using these metrics as predictors and the type of variation (inter, mean intra, std intra) as the predicted variable. For easier visualizuation, please note that we multiply all types of variation by 100, in both language emergence and language change scenarios. All predictors are z-scored.
We first look at the summary of each linear models.
Then, we gather the standardized estimates in a table.
We also use step models (using stats
package), which chooses a model by AIC in a stepwise Algorithm. The
Akaike information criterion (AIC) is an estimator of
prediction error and thereby relative quality of statistical models for
a given set of data. Given a collection of models for the data, AIC
estimates the quality of each model, relative to each of the other
models. Thus, AIC provides a means for model selection. Step models show
the best predictors and also indicates the most important predictors for
maximizing AIC (increasing importance). In other terms,
it shows how much the AIC increases when removing a predictor from the
model (the higher the AIC, the worst the model).
In order to look at the variable importance, we also integrate all variables in a model, then remove each of them to evaluate the decrease in the total R² of the model.
Finally, we create models with single predictors and evaluate the total R² that these predictors explain.
1 - Summary of the linear models.
For random networks, in language emergence:
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)
Residuals:
Min 1Q Median 3Q Max
-0.38396 -0.01526 0.00626 0.03998 0.59269
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.050242 0.005232 9.603 < 2e-16 ***
clustering_z 0.182628 0.014649 12.467 < 2e-16 ***
assortativity_z -0.015529 0.006194 -2.507 0.01271 *
pathlength_z 0.363731 0.009888 36.785 < 2e-16 ***
size_z -0.028448 0.011487 -2.477 0.01383 *
node_degree_z 0.039037 0.014224 2.744 0.00644 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.09046 on 293 degrees of freedom
Multiple R-squared: 0.8838, Adjusted R-squared: 0.8818
F-statistic: 445.7 on 5 and 293 DF, p-value: < 2.2e-16
For random networks, in language change:
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)
Residuals:
Min 1Q Median 3Q Max
-0.252490 -0.011758 0.004671 0.027376 0.292066
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.033849 0.003471 9.752 < 2e-16 ***
clustering_z 0.131623 0.009642 13.651 < 2e-16 ***
assortativity_z -0.015461 0.004422 -3.496 0.000545 ***
pathlength_z 0.250502 0.006404 39.120 < 2e-16 ***
size_z -0.010637 0.008076 -1.317 0.188841
node_degree_z 0.021007 0.009487 2.214 0.027587 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0599 on 292 degrees of freedom
Multiple R-squared: 0.8902, Adjusted R-squared: 0.8883
F-statistic: 473.3 on 5 and 292 DF, p-value: < 2.2e-16
For small-world networks, in language emergence:
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)
Residuals:
Min 1Q Median 3Q Max
-0.46288 -0.17581 -0.01208 0.13034 1.09315
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.35133 0.01400 25.103 < 2e-16 ***
clustering_z 0.22439 0.02085 10.762 < 2e-16 ***
assortativity_z 0.08090 0.01570 5.152 4.73e-07 ***
pathlength_z 0.68205 0.02354 28.972 < 2e-16 ***
size_z -0.11873 0.02010 -5.905 9.70e-09 ***
node_degree_z 0.04566 0.02653 1.721 0.0863 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2424 on 294 degrees of freedom
Multiple R-squared: 0.8479, Adjusted R-squared: 0.8453
F-statistic: 327.8 on 5 and 294 DF, p-value: < 2.2e-16
For small-world networks, in language change:
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)
Residuals:
Min 1Q Median 3Q Max
-0.30734 -0.11059 -0.00921 0.07822 1.00512
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.236805 0.009445 25.072 < 2e-16 ***
clustering_z 0.140855 0.014211 9.912 < 2e-16 ***
assortativity_z 0.044994 0.010591 4.248 2.89e-05 ***
pathlength_z 0.420955 0.016445 25.597 < 2e-16 ***
size_z -0.074567 0.013726 -5.432 1.17e-07 ***
node_degree_z 0.023018 0.018027 1.277 0.203
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1636 on 294 degrees of freedom
Multiple R-squared: 0.8197, Adjusted R-squared: 0.8167
F-statistic: 267.4 on 5 and 294 DF, p-value: < 2.2e-16
For scale-free networks, in language emergence:
Call:
lm(formula = inter_var * 100 ~ exponent_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sf_em)
Residuals:
Min 1Q Median 3Q Max
-1.47445 -0.33898 -0.01628 0.31736 1.61598
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.30238 0.03205 103.045 < 2e-16 ***
exponent_z -0.07074 0.04646 -1.523 0.12894
assortativity_z -0.01776 0.04736 -0.375 0.70791
pathlength_z 0.32245 0.05672 5.685 3.15e-08 ***
size_z 0.18763 0.06854 2.737 0.00657 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.5551 on 295 degrees of freedom
Multiple R-squared: 0.3867, Adjusted R-squared: 0.3784
F-statistic: 46.5 on 4 and 295 DF, p-value: < 2.2e-16
For scale-free networks, in language change:
Call:
lm(formula = inter_var * 100 ~ exponent_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sf_ch)
Residuals:
Min 1Q Median 3Q Max
-0.76479 -0.21379 -0.01904 0.18163 1.69505
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.035471 0.019081 106.673 < 2e-16 ***
exponent_z -0.002603 0.028557 -0.091 0.9274
assortativity_z 0.027438 0.027381 1.002 0.3171
pathlength_z 0.187230 0.032754 5.716 2.67e-08 ***
size_z 0.072220 0.042794 1.688 0.0925 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3305 on 295 degrees of freedom
Multiple R-squared: 0.3951, Adjusted R-squared: 0.3869
F-statistic: 48.18 on 4 and 295 DF, p-value: < 2.2e-16
2 - Summary of the standardized estimates of the linear models.
For easier visualization, the following table gathers the effect size, in absolute value, rounded and ordered from the highest to the lowest, for each network type.
In language emergence:
| metrics | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| (Intercept) | 0.05 | 0.35 | 3.30 |
| assortativity | 0.02 | 0.08 | 0.02 |
| clustering | 0.18 | 0.22 | NA |
| exponent | NA | NA | 0.07 |
| node degree | 0.04 | 0.05 | NA |
| pathlength | 0.36 | 0.68 | 0.32 |
| size | 0.03 | 0.12 | 0.19 |
In language change:
| metrics | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| (Intercept) | 0.03 | 0.24 | 2.04 |
| assortativity | 0.02 | 0.04 | 0.03 |
| clustering | 0.13 | 0.14 | NA |
| exponent | NA | NA | 0.00 |
| node degree | 0.02 | 0.02 | NA |
| pathlength | 0.25 | 0.42 | 0.19 |
| size | 0.01 | 0.07 | 0.07 |
3 - Step models (looking at AIC).
For random networks, in language emergence:
Start: AIC=-1430.97
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
<none> 2.3976 -1431.0
- size_z 1 0.0502 2.4478 -1426.8
- assortativity_z 1 0.0514 2.4490 -1426.6
- node_degree_z 1 0.0616 2.4592 -1425.4
- clustering_z 1 1.2718 3.6694 -1305.7
- pathlength_z 1 11.0723 13.4699 -916.9
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)
Coefficients:
(Intercept) clustering_z assortativity_z pathlength_z
0.05024 0.18263 -0.01553 0.36373
size_z node_degree_z
-0.02845 0.03904
For random networks, in language change:
Start: AIC=-1671.8
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- size_z 1 0.0062 1.0541 -1672.0
<none> 1.0479 -1671.8
- node_degree_z 1 0.0176 1.0654 -1668.8
- assortativity_z 1 0.0439 1.0917 -1661.6
- clustering_z 1 0.6688 1.7166 -1526.7
- pathlength_z 1 5.4917 6.5395 -1128.1
Step: AIC=-1672.04
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
node_degree_z
Df Sum of Sq RSS AIC
<none> 1.0541 -1672.0
- node_degree_z 1 0.0158 1.0699 -1669.6
- assortativity_z 1 0.0648 1.1189 -1656.3
- clustering_z 1 1.5305 2.5846 -1406.8
- pathlength_z 1 6.1628 7.2168 -1100.8
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + node_degree_z, data = subdf_mod_ran_ch)
Coefficients:
(Intercept) clustering_z assortativity_z pathlength_z
0.03380 0.14061 -0.01754 0.25304
node_degree_z
0.01031
For small-world networks, in language emergence:
Start: AIC=-844.33
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
<none> 17.276 -844.33
- node_degree_z 1 0.174 17.450 -843.33
- assortativity_z 1 1.560 18.836 -820.40
- size_z 1 2.049 19.326 -812.70
- clustering_z 1 6.806 24.082 -746.70
- pathlength_z 1 49.324 66.600 -441.52
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)
Coefficients:
(Intercept) clustering_z assortativity_z pathlength_z
0.35133 0.22439 0.08090 0.68205
size_z node_degree_z
-0.11873 0.04566
For small-world networks, in language change:
Start: AIC=-1080.3
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- node_degree_z 1 0.0436 7.9116 -1080.64
<none> 7.8679 -1080.30
- assortativity_z 1 0.4830 8.3510 -1064.42
- size_z 1 0.7897 8.6577 -1053.60
- clustering_z 1 2.6292 10.4971 -995.80
- pathlength_z 1 17.5349 25.4028 -730.68
Step: AIC=-1080.64
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
<none> 7.912 -1080.64
- assortativity_z 1 0.462 8.373 -1065.62
- size_z 1 0.889 8.800 -1050.70
- clustering_z 1 4.906 12.817 -937.90
- pathlength_z 1 34.134 42.046 -581.51
Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sw_ch)
Coefficients:
(Intercept) clustering_z assortativity_z pathlength_z
0.23681 0.15198 0.04382 0.40577
size_z
-0.06439
For scale-free networks, in language emergence:
Start: AIC=-348.22
inter_var * 100 ~ exponent_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
- assortativity_z 1 0.0433 90.938 -350.08
<none> 90.895 -348.22
- exponent_z 1 0.7143 91.609 -347.88
- size_z 1 2.3088 93.204 -342.70
- pathlength_z 1 9.9567 100.851 -319.04
Step: AIC=-350.08
inter_var * 100 ~ exponent_z + pathlength_z + size_z
Df Sum of Sq RSS AIC
<none> 90.938 -350.08
- exponent_z 1 0.8760 91.814 -349.20
- size_z 1 2.2863 93.224 -344.63
- pathlength_z 1 10.0827 101.021 -320.54
Call:
lm(formula = inter_var * 100 ~ exponent_z + pathlength_z + size_z,
data = subdf_mod_sf_em)
Coefficients:
(Intercept) exponent_z pathlength_z size_z
3.30238 -0.07544 0.31833 0.18174
For scale-free networks, in language change:
Start: AIC=-659.33
inter_var * 100 ~ exponent_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
- exponent_z 1 0.0009 32.224 -661.32
- assortativity_z 1 0.1097 32.333 -660.31
<none> 32.223 -659.33
- size_z 1 0.3111 32.534 -658.45
- pathlength_z 1 3.5691 35.792 -629.82
Step: AIC=-661.32
inter_var * 100 ~ assortativity_z + pathlength_z + size_z
Df Sum of Sq RSS AIC
- assortativity_z 1 0.1108 32.335 -662.29
<none> 32.224 -661.32
- size_z 1 0.4046 32.628 -659.58
- pathlength_z 1 3.8345 36.058 -629.59
Step: AIC=-662.29
inter_var * 100 ~ pathlength_z + size_z
Df Sum of Sq RSS AIC
<none> 32.335 -662.29
- size_z 1 0.8466 33.181 -656.54
- pathlength_z 1 3.8690 36.204 -630.39
Call:
lm(formula = inter_var * 100 ~ pathlength_z + size_z, data = subdf_mod_sf_ch)
Coefficients:
(Intercept) pathlength_z size_z
2.0355 0.1888 0.0883
4 - Drop in R² when removing predictors
Language emergence:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0616377 | 0.0599168 | NA |
| node_degree | 0.0029870 | 0.0015321 | NA |
| assortativity | 0.0024929 | 0.0137317 | 0.0002924 |
| pathlength | 0.5366329 | 0.4342481 | 0.0671823 |
| size | 0.0024325 | 0.0180421 | 0.0155788 |
| exponent | NA | NA | 0.0048196 |
Language change:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0701024 | 0.0602336 | NA |
| node_degree | 0.0018442 | 0.0009996 | NA |
| assortativity | 0.0045983 | 0.0110663 | 0.0020589 |
| pathlength | 0.5756553 | 0.4017201 | 0.0669968 |
| size | 0.0006525 | 0.0180929 | 0.0058397 |
| exponent | NA | NA | 0.0000170 |
5 - R² explained by each predictor
Language emergence:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0974584 | -0.0014516 | NA |
| node_degree | 0.0388764 | 0.1682322 | NA |
| assortativity | 0.1480515 | -0.0032237 | 0.1495115 |
| pathlength | 0.6182816 | 0.6401498 | 0.3687975 |
| size | 0.0413732 | 0.0136092 | 0.3018147 |
| exponent | NA | NA | 0.0824252 |
Language change:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0998072 | -0.0027987 | NA |
| node_degree | 0.0398155 | 0.1896333 | NA |
| assortativity | 0.1463084 | 0.0048742 | 0.1841091 |
| pathlength | 0.6218682 | 0.6223793 | 0.3750539 |
| size | 0.0418832 | 0.0092948 | 0.3181298 |
| exponent | NA | NA | 0.1261277 |
We apply here exactly the same steps, but using the mean intra-individual variation as the predicted variable. Please note that here too, for better visualization, we multiply the mean intra-individual variation by 100.
1 - Summary of the linear models.
For random networks, in language emergence:
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)
Residuals:
Min 1Q Median 3Q Max
-19.9060 -1.2994 0.2065 1.7039 7.5610
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 225.4411 0.1636 1377.875 < 2e-16 ***
clustering_z -1.5649 0.4581 -3.416 0.000726 ***
assortativity_z 0.3292 0.1937 1.700 0.090252 .
pathlength_z 0.1962 0.3092 0.635 0.526233
size_z 1.4270 0.3592 3.972 8.96e-05 ***
node_degree_z 1.2852 0.4448 2.889 0.004151 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.829 on 293 degrees of freedom
Multiple R-squared: 0.4707, Adjusted R-squared: 0.4617
F-statistic: 52.12 on 5 and 293 DF, p-value: < 2.2e-16
For random networks, in language change:
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)
Residuals:
Min 1Q Median 3Q Max
-35.987 -4.759 0.314 4.808 25.321
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 195.4888 0.4879 400.647 <2e-16 ***
clustering_z -2.8852 1.3554 -2.129 0.0341 *
assortativity_z 1.0242 0.6216 1.648 0.1005
pathlength_z 0.1762 0.9002 0.196 0.8450
size_z -0.5688 1.1353 -0.501 0.6167
node_degree_z 2.9409 1.3337 2.205 0.0282 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 8.421 on 292 degrees of freedom
Multiple R-squared: 0.08716, Adjusted R-squared: 0.07153
F-statistic: 5.576 on 5 and 292 DF, p-value: 6.343e-05
For small-world networks, in language emergence:
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)
Residuals:
Min 1Q Median 3Q Max
-12.7624 -0.8564 0.0747 1.2064 7.2658
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 226.0901 0.1415 1597.391 < 2e-16 ***
clustering_z -2.0967 0.2109 -9.944 < 2e-16 ***
assortativity_z 0.2081 0.1588 1.311 0.191
pathlength_z -0.3928 0.2381 -1.650 0.100
size_z 1.3657 0.2033 6.717 9.59e-11 ***
node_degree_z -0.1165 0.2683 -0.434 0.665
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.451 on 294 degrees of freedom
Multiple R-squared: 0.5619, Adjusted R-squared: 0.5545
F-statistic: 75.43 on 5 and 294 DF, p-value: < 2.2e-16
For small-world networks, in language change:
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)
Residuals:
Min 1Q Median 3Q Max
-28.4825 -3.5119 0.3649 4.2764 20.2090
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 195.7702 0.4018 487.277 < 2e-16 ***
clustering_z -1.6216 0.6045 -2.683 0.00772 **
assortativity_z 0.3445 0.4505 0.765 0.44509
pathlength_z -0.8917 0.6995 -1.275 0.20342
size_z 1.2261 0.5839 2.100 0.03659 *
node_degree_z -1.2524 0.7668 -1.633 0.10348
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 6.959 on 294 degrees of freedom
Multiple R-squared: 0.1209, Adjusted R-squared: 0.106
F-statistic: 8.09 on 5 and 294 DF, p-value: 3.618e-07
For scale-free networks, in language emergence:
Call:
lm(formula = mean_intra_var * 100 ~ exponent_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sf_em)
Residuals:
Min 1Q Median 3Q Max
-8.4006 -0.8030 0.0626 0.8368 3.8055
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 221.71443 0.09106 2434.878 <2e-16 ***
exponent_z 0.17717 0.13202 1.342 0.1806
assortativity_z 0.34523 0.13455 2.566 0.0108 *
pathlength_z 0.05753 0.16117 0.357 0.7214
size_z -0.17637 0.19476 -0.906 0.3659
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.577 on 295 degrees of freedom
Multiple R-squared: 0.05738, Adjusted R-squared: 0.0446
F-statistic: 4.489 on 4 and 295 DF, p-value: 0.001552
For scale-free networks, in language change:
Call:
lm(formula = mean_intra_var * 100 ~ exponent_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sf_ch)
Residuals:
Min 1Q Median 3Q Max
-14.2636 -2.5661 0.4168 2.8622 16.3981
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 192.97293 0.26333 732.805 <2e-16 ***
exponent_z 0.44996 0.39411 1.142 0.255
assortativity_z 0.07738 0.37787 0.205 0.838
pathlength_z 0.04983 0.45203 0.110 0.912
size_z -0.14192 0.59058 -0.240 0.810
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.561 on 295 degrees of freedom
Multiple R-squared: 0.008748, Adjusted R-squared: -0.004693
F-statistic: 0.6508 on 4 and 295 DF, p-value: 0.6267
2 - Summary of the standardized estimates of the linear models.
For easier visualization, the following table gathers the effect size, in absolute value, rounded and ordered from the highest to the lowest, for each network type.
In language emergence:
| metrics | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| (Intercept) | 225.44 | 226.09 | 221.71 |
| assortativity | 0.33 | 0.21 | 0.35 |
| clustering | 1.56 | 2.10 | NA |
| exponent | NA | NA | 0.18 |
| node degree | 1.29 | 0.12 | NA |
| pathlength | 0.20 | 0.39 | 0.06 |
| size | 1.43 | 1.37 | 0.18 |
In language change:
| metrics | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| (Intercept) | 195.49 | 195.77 | 192.97 |
| assortativity | 1.02 | 0.34 | 0.08 |
| clustering | 2.89 | 1.62 | NA |
| exponent | NA | NA | 0.45 |
| node degree | 2.94 | 1.25 | NA |
| pathlength | 0.18 | 0.89 | 0.05 |
| size | 0.57 | 1.23 | 0.14 |
3 - Step models (looking at AIC).
For random networks, in language emergence:
Start: AIC=627.81
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- pathlength_z 1 3.222 2348.1 626.22
<none> 2344.9 627.81
- assortativity_z 1 23.120 2368.0 628.74
- node_degree_z 1 66.804 2411.7 634.21
- clustering_z 1 93.381 2438.3 637.48
- size_z 1 126.290 2471.2 641.49
Step: AIC=626.22
mean_intra_var * 100 ~ clustering_z + assortativity_z + size_z +
node_degree_z
Df Sum of Sq RSS AIC
<none> 2348.1 626.22
- assortativity_z 1 20.005 2368.1 626.76
- node_degree_z 1 87.482 2435.6 635.16
- size_z 1 126.851 2475.0 639.95
- clustering_z 1 265.357 2613.5 656.23
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z +
size_z + node_degree_z, data = subdf_mod_ran_em)
Coefficients:
(Intercept) clustering_z assortativity_z size_z
225.4416 -1.7795 0.2913 1.3533
node_degree_z
1.3819
For random networks, in language change:
Start: AIC=1275.86
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- pathlength_z 1 2.72 20710 1273.9
- size_z 1 17.80 20725 1274.1
<none> 20707 1275.9
- assortativity_z 1 192.52 20900 1276.6
- clustering_z 1 321.34 21029 1278.5
- node_degree_z 1 344.83 21052 1278.8
Step: AIC=1273.9
mean_intra_var * 100 ~ clustering_z + assortativity_z + size_z +
node_degree_z
Df Sum of Sq RSS AIC
- size_z 1 24.46 20734 1272.2
<none> 20710 1273.9
- assortativity_z 1 190.87 20901 1274.6
- node_degree_z 1 406.48 21117 1277.7
- clustering_z 1 750.65 21461 1282.5
Step: AIC=1272.25
mean_intra_var * 100 ~ clustering_z + assortativity_z + node_degree_z
Df Sum of Sq RSS AIC
<none> 20734 1272.2
- assortativity_z 1 168.19 20903 1272.7
- node_degree_z 1 873.10 21608 1282.5
- clustering_z 1 1245.96 21980 1287.6
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z +
node_degree_z, data = subdf_mod_ran_ch)
Coefficients:
(Intercept) clustering_z assortativity_z node_degree_z
195.4880 -2.6645 0.8417 2.4009
For small-world networks, in language emergence:
Start: AIC=543.96
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- node_degree_z 1 1.13 1768.0 542.15
- assortativity_z 1 10.32 1777.2 543.71
<none> 1766.9 543.96
- pathlength_z 1 16.36 1783.2 544.72
- size_z 1 271.14 2038.0 584.79
- clustering_z 1 594.24 2361.1 628.93
Step: AIC=542.15
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
- assortativity_z 1 9.93 1778.0 541.83
<none> 1768.0 542.15
- pathlength_z 1 22.11 1790.1 543.88
- size_z 1 359.76 2127.8 595.72
- clustering_z 1 963.24 2731.3 670.62
Step: AIC=541.83
mean_intra_var * 100 ~ clustering_z + pathlength_z + size_z
Df Sum of Sq RSS AIC
<none> 1778.0 541.83
- pathlength_z 1 29.19 1807.2 544.72
- size_z 1 464.02 2242.0 609.40
- clustering_z 1 998.80 2776.8 673.58
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + pathlength_z +
size_z, data = subdf_mod_sw_em)
Coefficients:
(Intercept) clustering_z pathlength_z size_z
226.0901 -2.0836 -0.3596 1.3962
For small-world networks, in language change:
Start: AIC=1169.94
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- assortativity_z 1 28.31 14265 1168.5
- pathlength_z 1 78.68 14315 1169.6
<none> 14237 1169.9
- node_degree_z 1 129.17 14366 1170.7
- size_z 1 213.53 14450 1172.4
- clustering_z 1 348.48 14585 1175.2
Step: AIC=1168.53
mean_intra_var * 100 ~ clustering_z + pathlength_z + size_z +
node_degree_z
Df Sum of Sq RSS AIC
<none> 14265 1168.5
- pathlength_z 1 110.42 14375 1168.8
- node_degree_z 1 140.95 14406 1169.5
- size_z 1 292.89 14558 1172.6
- clustering_z 1 320.54 14586 1173.2
Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + pathlength_z +
size_z + node_degree_z, data = subdf_mod_sw_ch)
Coefficients:
(Intercept) clustering_z pathlength_z size_z node_degree_z
195.770 -1.505 -1.024 1.365 -1.303
For scale-free networks, in language emergence:
Start: AIC=278.34
mean_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
- pathlength_z 1 0.3170 734.12 276.46
- size_z 1 2.0400 735.84 277.17
- exponent_z 1 4.4797 738.28 278.16
<none> 733.80 278.34
- assortativity_z 1 16.3748 750.17 282.96
Step: AIC=276.47
mean_intra_var * 100 ~ exponent_z + assortativity_z + size_z
Df Sum of Sq RSS AIC
- size_z 1 2.0071 736.12 275.28
- exponent_z 1 4.1829 738.30 276.17
<none> 734.12 276.46
- assortativity_z 1 17.9453 752.06 281.71
Step: AIC=275.28
mean_intra_var * 100 ~ exponent_z + assortativity_z
Df Sum of Sq RSS AIC
- exponent_z 1 2.4603 738.58 274.29
<none> 736.12 275.28
- assortativity_z 1 16.5437 752.67 279.95
Step: AIC=274.29
mean_intra_var * 100 ~ assortativity_z
Df Sum of Sq RSS AIC
<none> 738.58 274.29
- assortativity_z 1 39.882 778.46 288.06
Call:
lm(formula = mean_intra_var * 100 ~ assortativity_z, data = subdf_mod_sf_em)
Coefficients:
(Intercept) assortativity_z
221.7144 0.3652
For scale-free networks, in language change:
Start: AIC=915.5
mean_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
- pathlength_z 1 0.2528 6137.3 913.51
- assortativity_z 1 0.8725 6137.9 913.54
- size_z 1 1.2013 6138.2 913.55
- exponent_z 1 27.1172 6164.2 914.82
<none> 6137.0 915.50
Step: AIC=913.51
mean_intra_var * 100 ~ exponent_z + assortativity_z + size_z
Df Sum of Sq RSS AIC
- assortativity_z 1 0.9543 6138.3 911.55
- size_z 1 1.0751 6138.4 911.56
- exponent_z 1 27.5329 6164.8 912.85
<none> 6137.3 913.51
Step: AIC=911.55
mean_intra_var * 100 ~ exponent_z + size_z
Df Sum of Sq RSS AIC
- size_z 1 0.4169 6138.7 909.57
- exponent_z 1 31.1564 6169.4 911.07
<none> 6138.3 911.55
Step: AIC=909.57
mean_intra_var * 100 ~ exponent_z
Df Sum of Sq RSS AIC
<none> 6138.7 909.57
- exponent_z 1 52.535 6191.2 910.13
Call:
lm(formula = mean_intra_var * 100 ~ exponent_z, data = subdf_mod_sf_ch)
Coefficients:
(Intercept) exponent_z
192.9729 0.4192
4 - Drop in R² when removing predictors
Language emergence:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0210765 | 0.1473304 | NA |
| node_degree | 0.0150780 | 0.0002807 | NA |
| assortativity | 0.0052184 | 0.0025592 | 0.0210348 |
| pathlength | 0.0007273 | 0.0040550 | 0.0004072 |
| size | 0.0285042 | 0.0672249 | 0.0026205 |
| exponent | NA | NA | 0.0057545 |
Language change:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0141656 | 0.0215168 | NA |
| node_degree | 0.0152013 | 0.0079759 | NA |
| assortativity | 0.0084867 | 0.0017482 | 0.0001409 |
| pathlength | 0.0001197 | 0.0048583 | 0.0000408 |
| size | 0.0007847 | 0.0131846 | 0.0001940 |
| exponent | NA | NA | 0.0043799 |
5 - R² explained by each predictor
Language emergence:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0508113 | 0.4421897 | NA |
| node_degree | 0.1074698 | 0.0737294 | NA |
| assortativity | 0.0964938 | -0.0029023 | 0.0480474 |
| pathlength | 0.0149308 | 0.0825762 | 0.0152155 |
| size | 0.3904400 | 0.3022928 | 0.0178461 |
| exponent | NA | NA | 0.0298955 |
Language change:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0109673 | 0.0977439 | NA |
| node_degree | 0.0120052 | 0.0382061 | NA |
| assortativity | 0.0237402 | -0.0031980 | 0.0003619 |
| pathlength | 0.0041205 | 0.0176431 | -0.0016285 |
| size | 0.0467455 | 0.0396394 | 0.0001765 |
| exponent | NA | NA | 0.0051582 |
We apply here exactly the same steps, but using the std intra-individual variation as the predicted variable. Please note that here too, for better visualizuation, we multiply the std intra-individual variation by 100.
1 - Summary of the linear models.
For random networks, in language emergence:
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)
Residuals:
Min 1Q Median 3Q Max
-1.4066 -0.0261 0.0054 0.0445 3.5336
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.17183 0.01723 9.974 < 2e-16 ***
clustering_z 0.28655 0.04824 5.940 8.05e-09 ***
assortativity_z 0.01291 0.02040 0.633 0.52712
pathlength_z 0.73946 0.03256 22.710 < 2e-16 ***
size_z -0.15570 0.03783 -4.116 5.01e-05 ***
node_degree_z 0.13964 0.04684 2.981 0.00311 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2979 on 293 degrees of freedom
Multiple R-squared: 0.7608, Adjusted R-squared: 0.7568
F-statistic: 186.4 on 5 and 293 DF, p-value: < 2.2e-16
For random networks, in language change:
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)
Residuals:
Min 1Q Median 3Q Max
-1.6454 -0.0497 0.0046 0.0599 3.2076
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.39832 0.01712 23.268 < 2e-16 ***
clustering_z 0.39675 0.04755 8.344 2.90e-15 ***
assortativity_z -0.14323 0.02181 -6.567 2.34e-10 ***
pathlength_z 1.20515 0.03158 38.160 < 2e-16 ***
size_z -0.15759 0.03983 -3.956 9.56e-05 ***
node_degree_z 0.19657 0.04679 4.201 3.53e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2954 on 292 degrees of freedom
Multiple R-squared: 0.9119, Adjusted R-squared: 0.9104
F-statistic: 604.4 on 5 and 292 DF, p-value: < 2.2e-16
For small-world networks, in language emergence:
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)
Residuals:
Min 1Q Median 3Q Max
-0.78311 -0.24866 -0.02859 0.17983 2.13887
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.660748 0.022323 29.599 < 2e-16 ***
clustering_z 0.236637 0.033257 7.115 8.55e-12 ***
assortativity_z 0.121258 0.025048 4.841 2.09e-06 ***
pathlength_z 0.861297 0.037550 22.937 < 2e-16 ***
size_z -0.243309 0.032068 -7.587 4.33e-13 ***
node_degree_z -0.005828 0.042319 -0.138 0.891
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3867 on 294 degrees of freedom
Multiple R-squared: 0.788, Adjusted R-squared: 0.7844
F-statistic: 218.5 on 5 and 294 DF, p-value: < 2.2e-16
For small-world networks, in language change:
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)
Residuals:
Min 1Q Median 3Q Max
-2.0754 -0.6464 -0.1776 0.3638 5.5210
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.12077 0.05996 35.372 < 2e-16 ***
clustering_z 0.68179 0.09021 7.558 5.24e-13 ***
assortativity_z 0.10574 0.06723 1.573 0.116824
pathlength_z 2.01130 0.10439 19.266 < 2e-16 ***
size_z -0.29570 0.08713 -3.394 0.000785 ***
node_degree_z -0.32307 0.11444 -2.823 0.005080 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.038 on 294 degrees of freedom
Multiple R-squared: 0.7779, Adjusted R-squared: 0.7741
F-statistic: 205.9 on 5 and 294 DF, p-value: < 2.2e-16
For scale-free networks, in language emergence:
Call:
lm(formula = std_intra_var * 100 ~ exponent_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sf_em)
Residuals:
Min 1Q Median 3Q Max
-2.2945 -0.5700 -0.1046 0.4263 4.7065
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.97639 0.05451 72.952 <2e-16 ***
exponent_z -0.07701 0.07903 -0.975 0.331
assortativity_z 0.11644 0.08054 1.446 0.149
pathlength_z 0.09859 0.09648 1.022 0.308
size_z 0.12200 0.11658 1.046 0.296
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9441 on 295 degrees of freedom
Multiple R-squared: 0.07062, Adjusted R-squared: 0.05802
F-statistic: 5.604 on 4 and 295 DF, p-value: 0.0002323
For scale-free networks, in language change:
Call:
lm(formula = std_intra_var * 100 ~ exponent_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sf_ch)
Residuals:
Min 1Q Median 3Q Max
-4.6536 -1.2839 -0.0560 0.9913 6.5168
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.71293 0.10685 81.541 <2e-16 ***
exponent_z 0.08114 0.15992 0.507 0.6123
assortativity_z 0.22253 0.15333 1.451 0.1478
pathlength_z 0.42153 0.18342 2.298 0.0222 *
size_z 0.14277 0.23964 0.596 0.5518
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.851 on 295 degrees of freedom
Multiple R-squared: 0.1445, Adjusted R-squared: 0.1329
F-statistic: 12.45 on 4 and 295 DF, p-value: 2.253e-09
2 - Summary of the standardized estimates of the linear models.
For easier visualization, the following table gathers the effect size, in absolute value, rounded and ordered from the highest to the lowest, for each network type.
In language emergence:
| metrics | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| (Intercept) | 0.17 | 0.66 | 3.98 |
| assortativity | 0.01 | 0.12 | 0.12 |
| clustering | 0.29 | 0.24 | NA |
| exponent | NA | NA | 0.08 |
| node degree | 0.14 | 0.01 | NA |
| pathlength | 0.74 | 0.86 | 0.10 |
| size | 0.16 | 0.24 | 0.12 |
In language change:
| metrics | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| (Intercept) | 0.40 | 2.12 | 8.71 |
| assortativity | 0.14 | 0.11 | 0.22 |
| clustering | 0.40 | 0.68 | NA |
| exponent | NA | NA | 0.08 |
| node degree | 0.20 | 0.32 | NA |
| pathlength | 1.21 | 2.01 | 0.42 |
| size | 0.16 | 0.30 | 0.14 |
3 - Step models (looking at AIC).
For random networks, in language emergence:
Start: AIC=-718.29
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- assortativity_z 1 0.036 26.033 -719.89
<none> 25.997 -718.29
- node_degree_z 1 0.789 26.786 -711.36
- size_z 1 1.503 27.501 -703.49
- clustering_z 1 3.131 29.128 -686.29
- pathlength_z 1 45.762 71.759 -416.71
Step: AIC=-719.89
std_intra_var * 100 ~ clustering_z + pathlength_z + size_z +
node_degree_z
Df Sum of Sq RSS AIC
<none> 26.033 -719.89
- node_degree_z 1 0.808 26.841 -712.75
- size_z 1 1.468 27.501 -705.49
- clustering_z 1 3.110 29.143 -688.14
- pathlength_z 1 49.713 75.746 -402.54
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + pathlength_z +
size_z + node_degree_z, data = subdf_mod_ran_em)
Coefficients:
(Intercept) clustering_z pathlength_z size_z node_degree_z
0.1718 0.2812 0.7331 -0.1519 0.1411
For random networks, in language change:
Start: AIC=-720.75
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
<none> 25.487 -720.75
- size_z 1 1.366 26.854 -707.19
- node_degree_z 1 1.541 27.028 -705.27
- assortativity_z 1 3.765 29.252 -681.70
- clustering_z 1 6.076 31.564 -659.03
- pathlength_z 1 127.107 152.594 -189.46
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)
Coefficients:
(Intercept) clustering_z assortativity_z pathlength_z
0.3983 0.3968 -0.1432 1.2052
size_z node_degree_z
-0.1576 0.1966
For small-world networks, in language emergence:
Start: AIC=-564.2
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
- node_degree_z 1 0.003 43.956 -566.18
<none> 43.953 -564.20
- assortativity_z 1 3.504 47.457 -543.19
- clustering_z 1 7.569 51.522 -518.53
- size_z 1 8.606 52.559 -512.55
- pathlength_z 1 78.656 122.609 -258.43
Step: AIC=-566.18
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
<none> 43.956 -566.18
- assortativity_z 1 3.505 47.461 -545.16
- clustering_z 1 11.393 55.349 -499.04
- size_z 1 12.515 56.471 -493.02
- pathlength_z 1 162.778 206.734 -103.70
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z, data = subdf_mod_sw_em)
Coefficients:
(Intercept) clustering_z assortativity_z pathlength_z
0.6607 0.2339 0.1210 0.8650
size_z
-0.2457
For small-world networks, in language change:
Start: AIC=28.59
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z +
size_z + node_degree_z
Df Sum of Sq RSS AIC
<none> 317.05 28.585
- assortativity_z 1 2.67 319.72 29.099
- node_degree_z 1 8.60 325.65 34.610
- size_z 1 12.42 329.47 38.112
- clustering_z 1 61.60 378.65 79.850
- pathlength_z 1 400.30 717.35 271.534
Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z +
pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)
Coefficients:
(Intercept) clustering_z assortativity_z pathlength_z
2.1208 0.6818 0.1057 2.0113
size_z node_degree_z
-0.2957 -0.3231
For scale-free networks, in language emergence:
Start: AIC=-29.56
std_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
- exponent_z 1 0.84646 263.78 -30.596
- pathlength_z 1 0.93082 263.87 -30.500
- size_z 1 0.97605 263.91 -30.448
<none> 262.94 -29.560
- assortativity_z 1 1.86287 264.80 -29.442
Step: AIC=-30.6
std_intra_var * 100 ~ assortativity_z + pathlength_z + size_z
Df Sum of Sq RSS AIC
- size_z 1 0.39653 264.18 -32.145
- assortativity_z 1 1.34494 265.13 -31.070
- pathlength_z 1 1.37558 265.16 -31.035
<none> 263.78 -30.596
Step: AIC=-32.15
std_intra_var * 100 ~ assortativity_z + pathlength_z
Df Sum of Sq RSS AIC
<none> 264.18 -32.145
- assortativity_z 1 2.4566 266.64 -31.368
- pathlength_z 1 4.5688 268.75 -29.001
Call:
lm(formula = std_intra_var * 100 ~ assortativity_z + pathlength_z,
data = subdf_mod_sf_em)
Coefficients:
(Intercept) assortativity_z pathlength_z
3.9764 0.1169 0.1595
For scale-free networks, in language change:
Start: AIC=374.31
std_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z +
size_z
Df Sum of Sq RSS AIC
- exponent_z 1 0.8817 1011.3 372.57
- size_z 1 1.2157 1011.7 372.67
<none> 1010.5 374.31
- assortativity_z 1 7.2145 1017.7 374.45
- pathlength_z 1 18.0913 1028.5 377.63
Step: AIC=372.57
std_intra_var * 100 ~ assortativity_z + pathlength_z + size_z
Df Sum of Sq RSS AIC
- size_z 1 3.5005 1014.8 371.61
<none> 1011.3 372.57
- assortativity_z 1 8.8381 1020.2 373.18
- pathlength_z 1 17.2246 1028.6 375.64
Step: AIC=371.61
std_intra_var * 100 ~ assortativity_z + pathlength_z
Df Sum of Sq RSS AIC
<none> 1014.8 371.61
- assortativity_z 1 20.061 1034.9 375.48
- pathlength_z 1 55.014 1069.8 385.45
Call:
lm(formula = std_intra_var * 100 ~ assortativity_z + pathlength_z,
data = subdf_mod_sf_ch)
Coefficients:
(Intercept) assortativity_z pathlength_z
8.7129 0.3145 0.5208
4 - Drop in R² when removing predictors
Language emergence:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0288024 | 0.0365109 | NA |
| node_degree | 0.0072549 | 0.0000137 | NA |
| assortativity | 0.0003272 | 0.0169003 | 0.0065845 |
| pathlength | 0.4209669 | 0.3794113 | 0.0032901 |
| size | 0.0138290 | 0.0415141 | 0.0034500 |
| exponent | NA | NA | 0.0029919 |
Language change:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.0210065 | 0.0431583 | NA |
| node_degree | 0.0053256 | 0.0060219 | NA |
| assortativity | 0.0130148 | 0.0018692 | 0.0061083 |
| pathlength | 0.4394101 | 0.2804588 | 0.0153175 |
| size | 0.0047232 | 0.0087014 | 0.0010293 |
| exponent | NA | NA | 0.0007465 |
5 - R² explained by each predictor
Language emergence:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.1228450 | -0.0031910 | NA |
| node_degree | 0.0625701 | 0.2353555 | NA |
| assortativity | 0.0986972 | -0.0024494 | 0.0468897 |
| pathlength | 0.5759031 | 0.6008074 | 0.0543806 |
| size | 0.0592300 | -0.0021660 | 0.0531710 |
| exponent | NA | NA | 0.0134618 |
Language change:
| term | Random | SmallWorld | ScaleFree |
|---|---|---|---|
| clustering | 0.2162100 | 0.0106864 | NA |
| node_degree | 0.1085096 | 0.3320156 | NA |
| assortativity | 0.2084149 | 0.0306273 | 0.0911472 |
| pathlength | 0.7483989 | 0.6690141 | 0.1208406 |
| size | 0.0678856 | 0.0096082 | 0.1181864 |
| exponent | NA | NA | 0.0629547 |
The models strongly suggest that pathlength is the most importance predictor predicting inter-individual variation, followed by the clustering coefficient.
The metrics do not predict well the mean intra-individual variation (R² too low).
For the standard-deviation of intra-individual, we also do find pathlength as the main predictor, followed by the clustering coefficient.
These results corrobate the one obtained with the different machine learning techniques. The only difference is that here, node degree is rarely an important predictor. This could be due to the fact that linear models do not handle well the high auto-correlation between the metrics (and node degree is strongly correlated to other metrics), while other machine learning are supposed to.
In this part of the paper, we investigate the influence of local network heterogeneity on language change.
To do so, the agents do not have the same initial Dirichlet distribution, but here, 20% of them are biased. A biased agents is an agent who has an initial Dirichlet distribution giving preference to utterance \(u_7\). These biased agents occupy nodes in the network based on their local metrics: for example, we bias the 20% agents with the highest or lowest node degree. These biased agents contrast with the unbiased agents, which either have no preference, either favor utterance \(u_4\) (both conditions for the initial language exposure variable: see Note on Initial language exposure terminology).
With this, we compare the effects of different local network metrics (see Table below), and we also compare these with resulting language in a set of networks with the same 20% of the agents are biased but placed in random nodes, as well as with homogeneous networks (i.e., where all nodes contain unbiased agents).
Why 20%? In our previous research (Josserand et al. (2021)), we investigated various other parameters. We encourage readers to refer to our previous paper if they wish to explore scenarios with different percentages of biased individuals in the population. In this study, we address a distinct research question: specifically, we examine the type of centrality most likely to result in high variation and widespread linguistic influence: is it betweenness centrality? eigenvector centrality? Or just people who have a lot of neighbors?
| Name metric | Description | Mathematical description |
|---|---|---|
| Node degree | Number of neighbors of a given node | Number of neighbors of a given node |
| Clustering coefficient | How well-connected are the neighbors of a given node (shows whether the node belongs to a clique) | The local clustering coefficient of a node is defined as: \(C_i = \frac{\lambda_G(v)}{\tau_G(v)}\), where \(\lambda_G(v)\) is the number of triangles on \(v \in V(G)\) for the undirected graph \(G\), and \(\tau_G(v)\) is the number of triples on \(v \in G\). The clustering coefficient varies between 0 (the friends of my friends are never connected) and 1 (the friends of my friends are all connected) |
| Betweenness centrality | Proportion of shrotest paths between members of the network that passes through a given node | Number of neighbors of a given node |
| Eigenvector centrality | The amount of influence a node has on a network: nodes that are connected to many other nodes that are themselves well-connected (and so on) get a higher Eigenvector centrality score | For a given graph \(G :=(V, E)\) with \(|V|\) the number of nodes and \(A = (a_{v,t})\) the adjacency matrix. The relative centrality of a node \(v\) is given by: \(x_v = \frac{1}{\lambda} \sum_{t\in M(v)}x_t = \frac{1}{\lambda} \sum_{t \in G}{}a_{v,t}x_t\) where \(M(v)\) is a set of the neighbors of \(v\) and \(\lambda\) is constant. In this implementation, the eigenvector centrality is normalized such that the highest eigenvector centrality a node can have is 1. This implementation is designed to agree with Gephi’s implementation out to at least 3 decimal places. |
| Closeness centrality | The normalized average length of the shortest path between the node and all other nodes in the graph | The normalized closeness centrality is defined here as \(C(v) = \frac{N - 1}{\sum_{u}{}d(u,v)}\), where \(d(u,v)\) is the distance between nodes \(u\) and \(v\), and \(N\) is the number of nodes |
Table 2. Table showing the different metrics used to describe the local structure of the network.
We study 4 different outputs:
Here too, we only look at multinomial language.
Before looking at our research questions, let’s just have a look at
the distribution of our micro-level metrics in networks (without
individual bias at birth). We want to observe how they are distributed
in order to check whether the 20% with the highest and lowest value are
equivalent to similar things. To do so, we generated other simulations
in which language does not evolve (cf, we just generated the networks
and recorded the micro-metrics for all nodes; see
DATASET3_Supplementary.csv).
For each network type, we observe the results for 50 simulations. Please note that we look at small-world network with a rewire probability = 0.1 and 0.9.
Checking the correlations: have agents with higher centrality measure also have more neighbors? To investigate this, we observe whether the node degree is correlated with the other centrality measures.
In random networks:
In small-world networks (rewire 0.1):
In small-world networks (rewire 0.0):
In scale-free networks:
We generated networks with biased and unbiased agents: the biased agents are the most popular (namely, the agents with the highest level of a certain metric) or the least popular (namely, the agents with the lowest level of a certain metric). We hope that using this method, the simulations have achieved the following:
And this should occur for all metrics. The table below helps us visualize the characteristics of the simulations:
| Type_Bias | Type_agents | Closeness_c | Eigenvector_c | Betweenness_c | Degree |
|---|---|---|---|---|---|
| Lowest | Biased | 0.1538371 | 0.0030046 | 0.00000 | 7.505222 |
| Lowest | Unbiased | 0.2190507 | 0.0882096 | 383.27344 | 13.001306 |
| Highest | Biased | 0.2683714 | 0.2306794 | 1415.11767 | 16.653556 |
| Highest | Unbiased | 0.1933236 | 0.0285095 | 26.84778 | 10.769056 |
Figure 57. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.
Figure 58. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).
Figure 59. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 60. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 61. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
| Type_Bias | Type_agents | Closeness_c | Eigenvector_c | Betweenness_c | Clustering | Degree |
|---|---|---|---|---|---|---|
| Lowest | Biased | 0.4052023 | 0.3170923 | 33.96828 | 0.0299558 | 7.505222 |
| Lowest | Unbiased | 0.4498472 | 0.5817441 | 110.42265 | 0.0923585 | 13.001306 |
| Highest | Biased | 0.4747078 | 0.7609381 | 177.14161 | 0.1363100 | 16.653556 |
| Highest | Unbiased | 0.4321785 | 0.4685659 | 74.55713 | 0.0663732 | 10.769056 |
Figure 62. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.
Figure 63. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).
Figure 64. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 65. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 66. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
| Type_Bias | Type_agents | Closeness_c | Eigenvector_c | Betweenness_c | Clustering | Degree |
|---|---|---|---|---|---|---|
| Lowest | Biased | 0.1498508 | 0.1526149 | 66.95883 | 0.1427405 | 3.196222 |
| Lowest | Unbiased | 0.1843173 | 0.3945869 | 422.85374 | 0.4367896 | 4.200944 |
| Highest | Biased | 0.2029817 | 0.6440126 | 874.74031 | 0.5578889 | 4.818333 |
| Highest | Unbiased | 0.1689271 | 0.2803554 | 222.05667 | 0.3295591 | 3.795417 |
Figure 67. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.
Figure 68. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).
Figure 69. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 70. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 71. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
| Type_Bias | Type_agents | Closeness_c | Eigenvector_c | Betweenness_c | Clustering | Degree |
|---|---|---|---|---|---|---|
| Lowest | Biased | 0.2383994 | 0.1565062 | 46.89362 | 0.0000000 | 2.338778 |
| Lowest | Unbiased | 0.2728753 | 0.3891900 | 247.19571 | 0.0267426 | 4.415306 |
| Highest | Biased | 0.2945326 | 0.6137915 | 449.37393 | 0.1095590 | 6.064778 |
| Highest | Unbiased | 0.2589567 | 0.2815458 | 146.75738 | 0.0001146 | 3.483806 |
Figure 72. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.
Figure 73. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).
Figure 74. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 75. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
Figure 76. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).
Uncomment below to see the code for the plots in the main paper.